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tomekkorbak

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[ "It came out yesterday. ", "It may have been overshadowed by the appearance of a certain… penis. ", "But, just like Batman: Damned #1, Return Of Wolverine #1 has sold out from Marvel and Diamond and so the publisher is sending it back for a second printing, probably with a cover reprinting the final page as some kind of spoiler as usual.", "\n\nExpect it to turn up in stores on October 24th. ", "Until then, here's a look at the covers of another recent sell-out from Marvel… Fantastic Four #2, Venom and Venom: First Host…" ]

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[ "Visually Stunning Weddings\n\nSonia Roselli Make-up Class\n\nSo did you get a big wad of cash for the holidays? ", "Or maybe one of those Visa gift cards? ", "If you did, well, come on over my new Bestie! ", "Just kidding. ", "But if you do need a great way to spend your money, why not head over to Sonia Roselli? ", "Book a private make-up lesson class with one of Sonia’s fabulous make-up artists and receive a free set of amazing professional make-up air brushes. ", "Air? ", "The brushes are made of some uber-soft hair creating an airbrush look.", "\n\nThe $375 luxe brushes are gifted with the purchase of a class for $95. ", "Tres chic and a very good deal indeed. ", "Email susan@soniaroselli.com for scheduling!" ]

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[ "The present invention relates to a tape threading arrangement for a VTR (video tape recorder), and more particularly to one that threads tape from a cassette around a drum with substantially a 360 degree wrap angle.", "\nIn helical scan video recording, a magnetic tape is drawn from a supply reel, wrapped around a drum having a magnetic recording head disposed therein, and then provided to a supply reel. ", "A 360 degree wrap angle allows the drum diameter to be one-half the size of a 180 degree wrap angle configuration for compactness. ", "If the one-half diameter 360 degree drum is rotated at twice the angular speed of the 180 degree drum the same head-to-tape speed is achieved as well as a one field per revolution, non-segmented picture. ", "To achieve further compactness, and also convenience of loading and removal, it is desirable to have the supply and take-up reels coaxially mounted one above the other in a cassette. ", "However, it is difficult to provide automatic threading of tape from a cassette with a 360 degree wrap angle around a drum.", "\nOne solution is shown in U. S. Pat. ", "No. ", "4,191,979. ", "As shown therein, a pair of kidney-shaped \"grippers\" are used to achieve the threading. ", "However, the grippers are relatively long, and therefore when in their initial position, which is longitudinally disposed between the reels and the drum, a relatively large space must be provided for them. ", "This is done by having a relatively large space between the reels and the drum. ", "For certain applications, such as a VTR used with an ENG (electronic news gathering) camera, this space may be too large.", "\nIt is therefore desirable to provide a compact threading arrangement for a 360 degree wrap angle of tape from a cassette." ]

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[ "“Swedish Days” in Paris\n\nPrincess Victoria of Sweden attended the ‘Swedish Days’ in Paris on November 30th. ", "The event was organized by the Swedish authorities in Paris due to the rotating presidency of the European Union, which is currently held by the Scandinavian country.", "\n\nView the image at Svenskdam\n\nVictoria illuminated two Swedish christmas trees that were donated to the city of Paris. ", "They are placed outside the Hótel de Ville. ", "Deputy Mayor of Paris, Anne Hidalgo, the head of International Relations Mayor of the French capital, Pierre Schapira, and the Swedish Ambassador in France, Gunnar Lund also attended the event.", "\n\nAfterwards, Victoria’s program continued with a concert of the choir of the church of Sweden, on the occasion of the feast of Santa Lucia – an old Swedish tradition.", "\n\nIn the evening the Princess attended a gala dinnerat “Le Petit Palais”, organized by the authorities of Paris and the Institute of Foreign Trade of Sweden." ]

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[ "Known to be close to his family, DiCaprio has regularly invited his mom Irmelin to red carpet events and film afterparties — like this instance in 1996, when she accompanied her son to a Total Eclipse event." ]

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[ "Faustovirus\n\nFaustovirus is a genus of giant virus which infects amoebae associated with humans. ", "The virus was first isolated in 2015 and shown to be around 0.2 micrometers in diameter with a double stranded DNA genome of 466 kilobases predicted to encode 451 proteins. ", "Although classified as a nucleocytoplasmic large DNA virus (NCDLV), faustoviruses share less than a quarter of their genes with other NCLDVs; however, ~46% are homologous to bacterial genes and the remainder are orphan genes (ORFans). ", "Specifically, the gene encoding the major capsid protein (MCP) of faustovirus is different than that of its most closely related giant virus, asfivirus, as well as other NCLDVs. ", "In asfivirus, the gene encoding MCP is a single genomic fragment of ~2000 base pairs (bp), however, in faustovirus the MCP is encoded by 13 exons separated by 12 large introns. ", "The exons have a mean length of 149 bp and the introns have a mean length of 1,273 bp. ", "The presence of introns in faustovirus genes is highly unusual for viruses .", "\n\nReplication\t\nThe replication strategy of faustovirus in amoeba is similar to that of mimivirus. ", "Lasting 18 to 20 hours, the replication cycle begins with the amoeba ingesting individual viral particles through a process known as phagocytosis. ", "After about 2 to 4 hours post infection, virus particles are internalized via phagocytic vacuoles and are detected by the host. ", "While the particles appear near the host’s nucleus, there is no evidence that the virus is within the nucleus nor has an interaction with the nuclear membrane. ", "Similar to the mimivirus, in which a channel is created for particle proteins and DNA to travel though, the faustovirus particles empty their internal compartments into the amoeba’s cytoplasm. ", "In both viruses, the fusion leads to an eclipse phase in which the contents of particles become invisible inside the cytoplasm of the host. ", "However, the eclipse phase of the faustovirus is longer than the mimivirus, taking place from 4 to 6 hours post infection. ", "Characterized by a loss of its spherical shape and a decrease in surface area, the amoeba host cell undergoes reorganization, such that at 8 to 10 hours post infection there are new particles in a region forming a donut shape. ", "This region is the viral factory; it is distinct from the nucleus and is surrounded by mitochondria. ", "Between 12- and 18-hours post infection, the virus factory takes up the entirety of the cytoplasm, which is completely filled with new viral particles. ", "At 18- to 20-hours post infection, the viral particles are released through cell lysis .", "\n\nPathogenicity\t\nFaustovirus affects amoeba associated with the human environment, like Vermamoeba vermiformis; this particular amoeba has been found in hospital water networks, drinking water, human stool samples, and contact lenses of keratitis patients, thus it may be a possible carrying agent for viruses. ", "Faustoviruses have been found in sewage water from various geographical locations, such as Senegal, France, Lebanon, and Saudi Arabia. ", "Isolated strains of the virus have been detected in rodents, cattle, febrile and healthy humans, and well water and rivers. ", "Although faustovirus was found in humans, it is unknown whether it has a pathogenic effect on humans; more research is required to determine the mode of infection and consequences of infection, if any exist.", "\n\nReferences\n\nCategory:Nucleocytoplasmic large DNA viruses\nCategory:Asfarviridae\nCategory:Virus stubs" ]

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[ "Q:\n\nHow to rewrite Jasmine 2.0 custom matcher to use with Angular 1.5?", "\n\nAngularJS tutorial has custom matcher that is not working in Jasmine starting from version 2.0:\nbeforeEach(function(){\n this.addMatchers({\n toEqualData: function(expected) {\n return angular.equals(this.actual, expected);\n }\n });\n });\n\nThe attempt to modify matcher fails with error:\nTypeError: undefined is not an object (evaluating 'matcherCompare.apply')\nMy matcher definition:\nbeforeEach(function(){\n jasmine.addMatchers({\n toEqualData: function(util, customEqualityTesters, actual, expected) {\n return angular.equals(actual, expected);\n }\n });\n });\n\nJasmine 2.0 custom matcher docs.", "\nHow to make it work?", "\n\nA:\n\nbeforeEach(function () {\n jasmine.addMatchers({\n toEqualData: function () {\n return {\n compare: function (actual, expected) {\n return {pass: angular.equals(actual, expected)};\n }\n };\n }\n });\n });\n\n" ]

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[ "Thoracic Idiopathic Scoliosis Severity Is Highly Correlated with 3D Measures of Thoracic Kyphosis.", "\nLoss of thoracic kyphosis has been associated with thoracic idiopathic scoliosis. ", "Modern 3-dimensional (3D) imaging systems allow more accurate characterization of the scoliotic deformity than traditional radiographs. ", "In this study, we utilized 3D calculations to characterize the association between increasing scoliosis severity and changes in the sagittal and axial planes. ", "Patients evaluated in a scoliosis clinic and determined to have either a normal spine or idiopathic scoliosis were included in the analysis. ", "All underwent upright, biplanar radiography with 3D reconstructions. ", "Two-dimensional (2D) measurements of the magnitude of the thoracic major curve and the thoracic kyphosis were recorded. ", "Image processing and MATLAB analysis were utilized to produce a 3D calculation of thoracic kyphosis and apical vertebral axial rotation. ", "Regression analysis was performed to determine the correlation of 2D kyphosis, 3D kyphosis, and apical axial rotation with the magnitude of the thoracic major curve. ", "The 442 patients for whom 2D and 3D data were collected had a main thoracic curve magnitude ranging from 1° to 118°. ", "Linear regression analysis of the 2D and 3D T5-T12 kyphosis versus main thoracic curve magnitude yielded significant models (p < 0.05). ", "The 2D model had a minimally negative slope (-0.07), a small R value (0.02), and a poor correlation coefficient (-0.14). ", "In contrast, the 3D model had a strongly negative slope (-0.54), a high R value (0.56), and a strong correlation coefficient (-0.75). ", "Curve magnitude also had a strong correlation with loss of 3D T1-T12 kyphosis and increasing apical axial rotation. ", "Segmentally calculated 3D thoracic kyphosis had a strongly negative correlation with the magnitude of the main thoracic curve. ", "With near uniformity, 3D thoracic kyphosis progressively decreased as scoliosis magnitude increased, at a rate of more than half the increase in the main thoracic curve magnitude. ", "Analysis confirmed a surprisingly strong correlation between scoliosis severity and loss of 3D kyphosis that was absent in the 2D analysis. ", "A similarly strong correlation between curve magnitude and apical axial rotation was evident. ", "These findings lend further credence to the concept that scoliosis progresses in the coronal, sagittal, and axial planes simultaneously. ", "The findings of this study suggest that 3D assessment is critical for adequate characterization of the multiplanar deformity of idiopathic scoliosis and deformity in the sagittal plane is linked to deformity in the coronal plane. ", "Increasing severity of coronal plane curvature is associated with a progressive loss of thoracic kyphosis that should be anticipated so that the appropriate intraoperative techniques for correction of idiopathic scoliosis can be applied in all 3 planes." ]

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[ "Dayton Ballet\n\nThe Dayton Ballet is a ballet company based in Dayton, Ohio.", "\n\nIt was founded in 1937, making it the second oldest regional ballet company in the United States. ", "It is also called the \"Company of Premieres\" as it is committed to presenting new works, including a new full-length ballet every two years. ", "Dayton Ballet seasons typically comprises four works—familiar traditional ballets, the classic, family-friendly holiday staple The Nutcracker, as well as new and innovative works—and over 40 performances.", "\n\nThe Dayton Ballet performs in the historic 1,139-seat Victoria Theatre and the state-of-the-art, 2,300-seat Schuster Center.", "\n\nPart of The Dayton Performing Arts Alliance, which also oversees the Dayton Opera and the Dayton Philharmonic, the Ballet receives administrative and operational leadership and support from its parent organization. ", "Karen Russo Burke has been the Artistic Director of the Dayton Ballet since 2011.", "\n\nHistory\n\nDayton Ballet had its beginning when Josephine (Jo) Schwarz and her sister Hermene opened The Schwarz School of Dance in 1927. ", "Jo Schwarz later studied ballet and danced in Chicago, in New York at the School of American Ballet, and in Europe. ", "She danced on Broadway, but was forced to return home to Dayton after being injured while performing there. ", "In May 1938, Jo and Hermene gathered together the school's finest dancers, named the troupe \"The Experimental Group for Young Dancers,\" and staged a performance at the Dayton Art Institute. ", "This was the first performance of what is now the Dayton Ballet.", "\n\nJo was a pioneer of the American regional ballet movement of the mid-20th century. ", "Through years of persistence, she made Dayton a center of dance. ", "In 1958, the company restructured as the Dayton Civic Ballet, with a board of directors, and federal tax-exempt status. ", "In 1959, the Dayton Civic Ballet became a chartered member of the Northeast Regional Ballet Association. ", "The Schwarz sisters organized many regional dance festivals and choreography conferences. ", "In 1978, the company dropped the \"Civic\" designation and became the fully professional Dayton Ballet.", "\n\n1980–1990\nStuart Sebastian, a student of Josephine and Hermene Schwarz, assumed directorship of the company in 1980 at the invitation of Josephine Schwarz. ", "He had danced professionally for the Dayton Ballet and the National Ballet of Washington before assuming the role. ", "He had also choreographed in New York, Germany and England. ", " Sebastian led the Dayton Ballet for 10 years, in which time the company rose in stature and status. ", "After watching the company in 1981, dance critic Walter Terry wrote in Dance Magazine, \"In just one year the Dayton Ballet has moved from first-rate amateur rank into the category of professional ballet. ", "Of particular importance is the stature of the new choreography on view in Dayton. ", "Good dancers are now numerous; gifted choreographers remain a rare species. ", "Stuart Sebastian is one of this special breed.\"", "\n\nSebastian brought in new dancers and created the company's first full-length ballet, The Sleeping Beauty. ", " He choreographed over 25 new works. ", " Of those, six were full-length ballets, including Swan Lake and Dracula. ", "Under Sebastian, the Dayton Ballet toured more than 75 cities and took its first international tour to Jerash, Jordan. ", " In 1988, the company appeared on national television while performing in the opening ceremonies of the Pan American Games.", "\n\nTransition: 1991-1993\nFollowing the departure of Stuart Sebastian in 1990 and his subsequent death in January 1991 after a lengthy battly with AIDS, the company entered a period of transition. ", "James Clouser, former artistic director of the Houston Ballet, was brought in on a three year contract. ", "It was a bumpy and tumultuous time for the company, and a time of considerable change. ", "Clouser was the first outside, non-Daytonian director, the Dayton Ballet had ever seen, and was a considerable break from the company's past directors. ", "Many dancers and staff from the previous era left, and many new dancers and staff were hired. ", "According to Dayton Daily News articles written during the transition, the board of trustees was looking for change and that is why they brought in someone from the outside. ", "After three years of internal \"war\", as stated by the Dayton Daily News, in the fall of 1992 an executive director was brought in for the very first time in the company's history: Dermot Burke.", "\n\nRecent history: 1993–2012\nDermot Burke was a star principal dancer with the Joffrey Ballet in New York and had been artistic director of the American Repertory Ballet in New Jersey for 10 years. ", "He knew and choreographed for Sebastian and the Dayton Ballet back in 1984. ", "That association, in addition to his track record of success leading the company in New Jersey, made him the right person to move the company forward. ", "In the Fall of 1993 Dermot assumed the dual role of executive and artistic director of the company.", "\n\nUnder Dermot Burke, the company took on a more American flavor in the tradition of the Joffrey Ballet, embracing distinctly American dance literature, dancers and choreographers. ", "There was also a shift to a \"repertory company with lots of choreographic voices,\" as Burke stated. ", "This repertory includes and has held onto the work of Sebastian and the company's past, while at the same time including the work of outside choreographers.", "\n\n2012 merger\nFollowing more than two years of planning, the Dayton Ballet, Dayton Opera, and Dayton Philharmonic Orchestra—Dayton’s three classical performing arts organizations—became a new, single entity on July 1, 2012. ", "This new organization was named the Dayton Performing Arts Alliance. ", "It is the largest performing arts organization in the community. ", "Dayton Performing Arts Alliance performances are made possible in part by Montgomery County and Culture Works. ", "The organization also receives partial funding from the Ohio Arts Council.", "\n\n“It took great foresight, courage, and leadership to engage in the complex due diligence process and pursue a common vision. ", "We are thrilled with what this first-in-the-nation merger will mean for Greater Dayton,” Mike Parks, president of The Dayton Foundation, said of the alliance.", "\n\nThe Dayton Daily News called the merger \"a new day for arts in Dayton.”", "\n\nThe new organization has one administrative and operational core with one CEO and executive director, with each performing arts unit retaining its own artistic director.", "\n\nDayton Ballet School\nThe company's dependent dance school, Dayton Ballet School, is the oldest dance school in Dayton and one of the oldest in the US. ", "It is the only school in the Miami Valley that is linked to a professional dance company. ", "The mission of the Dayton Ballet School is \"to provide the very finest dance training to any child or adult who seeks to experience the wonder and the fun of dance.\"", "\n\nThe goal of the School is to serve the community by not only strengthening legs and arms, but also by introducing accomplishments and discipline to the spirit of all participants.", "\n\nAlumni\n Aaron Berenson, dancer with Dayton Ballet.", "\n Jeraldyne Blunden, founder and former director of the Dayton Contemporary Dance Company.", "\n Rachel Carmazzi, dancer with Dayton Ballet.", "\n Rebecca Carmazzi, dancer with Dayton Ballet, BalletMet, and now the North Carolina Dance Theatre.", "\n Dan Duell, dancer with the New York City Ballet and artistic director of Ballet Chicago.", "\n Joseph Duell, dancer with the New York City Ballet.", "\n Robert Eberly III, dancer with Dayton Ballet, Ballet West, and Boston Ballet.", "\n Penny Freeh, choreographer for the James Sewell Ballet.", "\n Jeff Gribler, dancer and ballet master with the Pennsylvania Ballet.", "\n Carol Jean Heller, Dayton Ballet School Director and dancer with Ballet Repertory in New York.", "\n Peter Means, dancer with Milwaukee Ballet, Washington Ballet and Ballet Met.", "\n Peter LeBreton Merz, dancer with Louisville Ballet and Cincinnati Ballet, Director of the Academy at Ballet West in Salt Lake City, UT.", "\n Cheryl Mrozowski, director of the Dance Company at Wheaton College (Massachusetts).", "\n Bonnie Pickard, dancer with Suzanne Farrell Ballet and North Carolina Dance Theatre.", "\n Jennifer Schildknecht, dancer with Dayton Ballet and Ballet West.", "\n Stuart Sebastian, former Dayton Ballet Artistic Director and dancer with National Ballet of Washington.", "\n Rosalee Calvert, dancer with \"Dayton Civic Ballet\" Dayton Ballet.", "\n Dustin Shane, dancer with San Francisco Ballet.", "\n Gabrielle Sharp, dancer with the Dayton Ballet.", "\n Jennifer Sydor, dancer with Kim Robards Dance.", "\n Jonothan Tabbert, dancer with the Charleston Ballet.", "\n Donna Wood, dancer with Alvin Ailey.", "\n Rebecca Wright, dancer with American Ballet Theatre, on Broadway in Merlin, and with the Joffrey Ballet.", "\n\nSee also\nDayton Philharmonic Orchestra\nDayton Opera\n\nReferences\n\nExternal links\nDayton Ballet, official website\n Dayton Ballet Company Collection, Special Collections and Archives, Wright State University, Dayton, OH\n\nCategory:Ballet companies in the United States\nCategory:Dance schools in the United States\nCategory:Culture of Dayton, Ohio\nCategory:Companies based in Dayton, Ohio\nCategory:1937 establishments in Ohio\nCategory:Tourist attractions in Dayton, Ohio\nCategory:Performing groups established in 1937\nCategory:Dance in Ohio" ]

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[ "(1) Field of the Invention\nThis invention relates in general to earth boring drill pipe and to hardsurfacing of tool joints or connection members relative to the drill pipe.", "\n(2) Description of the Prior Art\nThe most common drill pipe used in earth boring operations have connection members or tool joints on each end that are larger in diameter than the drill pipe. ", "Annular bands of hardfacing are commonly deposited on each tool joint. ", "One type of hardfacing has macroscopic sintered tungsten carbide granules within an alloy steel matrix. ", "Sintered tungsten carbide granules, as explained in U.S. Pat. ", "No. ", "3,800,891, comprise microscopic grains of tungsten carbide held together by a binder of an iron group metal, usually cobalt. ", "Sintered tungsten carbide hardfacing is normally applied on tool joints by rotating the tool joint, providing an arc with a consumable steel wire, discharging an inert gas around the wire, and gravity feeding sintered tungsten carbide granules into the weld puddle behind the wire.", "\nOne disadvantage of the resulting sintered tungsten carbide hardfacing is that many of the granules remain only partially embedded in the matrix, giving a rough, abrasive exterior. ", "In deep wells, intermediate strings of casing are set as the well is drilled. ", "While drilling deeper through a string of intermediate casing, the rough surface of the hardfacing can abrade and damage the casing. ", "Consequently, it is advantageous to have a hardfacing surface free of protruding tungsten carbide granules. ", "In U.S. Pat. ", "No. ", "4,243,727, entitled \"SURFACE SMOOTHED TOOL JOINT HARDFACING\", a method and apparatus is shown for hardfacing tool joints using tungsten carbide granules that results in a smooth exterior. ", "The granules are dropped directly into the arc, rather than behind the arc. ", "This concentrates the granules toward the bottom of the hardfacing deposit and results in a harder matrix near the bottom, as well.", "\nIn U.S. Pat. ", "No. ", "4,228,339, an improved method for hardsurfacing the tool joints is provided in which annular bands of hardfacing are applied using tungsten carbide granules in a steel matrix applied by means of an electrode to the welding puddle.", "\nIt has been discovered that it is extremely difficult to machine down the hardfacing containing the tungsten carbide granules, because of their extreme hardness. ", "The removal of the outer layer of tungsten carbide granules in hardfacing is necessary so that the hardfacing itself will not cut into or wear a surface of the casing through which the tool joint typically will be inserted. ", "In instances in which tungsten carbide granules are within the matrix forming the hardfacing, considerable time is required to machine down the tungsten carbide granules, thus contributing significantly to the total cost of the hardfacing operation. ", "The present invention is an improvement upon the process and product of the '339 patent, cited above, in that the puddle containing the steel matrix and the tungsten carbide granules therein is deposited in the grooves by the electrode to a given depth and thereafter the feed of granules into the weld puddle is discontinued with a continuation of the application of the puddle by the electrode to the groove to provide a protruding substantially tungsten carbide free hardfacing surface above the outer diameter of the tool joint. ", "The protruding substantially tungsten carbide free hardfacing may then be machined down to substantially the outer diameter of the tool joint. ", "Because the protruding hardfacing surface does not contain tungsten carbide granules, the machining operation may be effected with considerable savings of time and equipment.", "\nThe present invention provides a method of applying grooved bands of hardfacing on a tool joint for earth boring drill pipe, the hardfacing having tungsten carbide granules in an alloy steel matrix. ", "The method includes the steps of rotating the tool joint, providing an arc between an electrode and the tool joint to create a weld puddle. ", "The electrode is reciprocated parallel to the tool joint axis and granules are fed into the weld puddle. ", "The weld puddle with the granules is applied within the grooved bands around the exterior of the tool joint to a grooved depth of from between about 0.75 inches to about 0.150 inches. ", "Thereafter, the feed of the granules into the weld puddle is discontinued. ", "The weld puddle is continued to be applied around the tool joint and in the grooved bands to provide a protruding substantially tungsten carbide free hardfacing surface above the outer diameter of the tool joint of from between about 0.050 inches to about 0.175 inches. ", "Thereafter, the hardfacing surface is machined down to substantially the outer diameter of the tool joint." ]

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[ "Følg RA Sporten på Facebook !", "\n\nDe mørkeblå fikk mandag klarsignalet de har ventet på fra Sunderland.", "\n\nDen unge midtstopperen kommer til Viking på lån fram til sommeren.", "\n\nLES OGSÅ: Håper på Ledger-svar mandag\n\nDermed har endelig Viking-trener Ian Burchnall fått konkurransen han har ønsket seg i den utsatte posisjonen. ", "Viking har kun Karol Mets og Rasmus Martinsen som rene stoppere fra før.", "\n\n– Alt er nå i orden og vi jobber med å få ham på et fly hit tirsdag, sa Burchnall til RA mandag ettermiddag.", "\n\nKan debutere fredag\n\nStopperen skal mandag kveld spille for Sunderland U23 mot Tottenham U23. ", "Planen er å la ham få debutere for Viking så fort som overhodet mulig.", "\n\n– Ja. ", "Jeg håper å gi ham debuten mot FK Haugesund på fredag. ", "Vi får ta en titt på tilstanden hans etter kampen mot Tottenham, Forhåpentlig trener han som vanlig med oss på torsdag og spiller fredag, sier Viking-treneren.", "\n\nLES OGSÅ: Viking imponerte mot Jerv\n\nLeter videre i England\n\nBurchnall har lenge vært klar på at låneavtaler ikke er ideelle, men situasjonen er i ferd med å bli såpass prekær at han er villig til å gamble litt på et ubeskrevet britisk blad.", "\n\nEttersom klubben ikke har penger til å handle, fortalte han RA forrige uke at de mørkeblå også jobber videre med et par andre konkrete navn fra England som kan komme inn på lån.", "\n\nDet kan også godt være snakk om en offensiv spiller.", "\n\n– Her er det ingen utvikling enda, men vi håper å vite mer senere denne uken. ", "Det tar litt mer tid å få tak i spillere som ikke koster en krone. ", "Vi får prøve å få til litt magi, sier Burchnall til RA." ]

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[ "Tuesday, July 23, 2013\n\nTorque Mobile Quad core phone Droidz Quad price and specs\n\nTorque came out to offer their\nlatest quad core Android smartphone called Droidz Quad. ", "If Cherry Mobile,\nMyPhone and Starmobile has quad core phones on their mobile list, Torque also\noffer you their quad core phone at affordable price. ", "This phone comes with an\nAndroid Jellybean OS out in the box for you to enjoy thousand compatible games\nand apps you can install on this phone.", "\n\nDroid Quad sports a MediaTek\nMT6589 1.2GHz processor with 4GB internal storage and 512MB RAM which I think\nnot a good move in terms of specs competition. ", "This handset price is only 5999\npesos plus you can get a free screen protector and jelly case. ", "Torque Mobile\nDroidz Quad camera takes an 8 megapixels photos and VGA for video calling.", "\n\nTorque Mobile Droidz Quad full\nspecifications\n\nAndroid 4.2 Jellybean OS\n\n4.5 inch FWVGA capcitive touch\nscreen\n\n854x480 resolution\n\n8 megapixels rear camera with LED\nFlash\n\nVGA front cam\n\nMulti touch\n\n1.2Ghz MT6589 quad core processor\n\n512MB RAM\n\n4GB internal storage\n\nsupports microSD up tp 32GB\n\n3G / Wi-Fi / Bluetooth / Wi-Fi\nhotspot / GPS\n\nHSDPA 7.2Mbps/ HSUPA 5.76 Mbps\n\nFM Radio\n\n3-axis Gyro / Accelerometer\n\nPrice: 5999 pesos\n\nTo avail Torque Mobile Droidz Quad, just visit the nearest Torque stores and kiosks on you area." ]

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[ "Selma (2014) the movie is out. ", "Snowstorm barrels into the Northeast. ", "And the U.S. is not expected to fault Darrel Wilson, the Ferguson police officer that killed Michael Brown last August. ", "From the Feds’ decision, it is fair to assume that there could be some similarities in the findings of the Feds and that of the St Louis County, Missouri on what actually happened in the altercation between Michael Brown and Officer Wilson. ", "At the mouth of two witnesses, or at the mouth of three witnesses, shall a matter be established. ", "That is justice. ", "And that is a closed chapter.", "\n\nMeanwhile, once upon a time in the animal kingdom, a day came when all animals were reporting to the town hall for a general meeting. ", "But the hen, which felt it had more ‘important’ things to do, absented herself from the meeting but pledged to abide by every decision reached at the meeting by those who attended the meeting. ", "At the meeting, a decision was reached that a ceremony will take place and that the hen and her chicks will be killed to provide the meat for refreshment at the ceremony. ", "The hen wasn’t present to object to that decision because she refused to attend the meeting. ", "And as she pledged, she had to abide by the decision reached by others at the meeting.", "\n\nNow, when you visit most of the think tanks in the District of Columbia, New York City, Chicago, and California, you will discover that the presence of African-Americans are close to non-existent in our country’s apex policy-making institutions where policies that govern this country are hatched. ", "Some pockets of the Black people you will see walking the ‘corridors of power’ are mostly interns from Africa who are studying government-related courses here in the U.S.\n\nSame goes for all the departments of the federal government at the levels of relevance where folks effect policy decisions. ", "Most workers of African-American derivation you will see at these federal government agencies work administrative and general services jobs. ", "And not top level positions where policies that govern the domestic politics of the United States are formulated. ", "And same also go for state and local governments. ", "And that, my people is a sad and hard fact.", "\n\nWhy so?", "\n\nIn most African-Americans homes, because we started late in the pursuit of happyness as a result of slavery and racism, most parents, in order to escape poverty and score on social mobility, are more interested in channeling and nurturing their children’s interests in sports and entertainment rather than in public service. ", "This phenomenon is easily understood because a pro-football or a pro-basketball player in a family is a guaranteed meal ticket and a move away from poverty. ", "When people are disadvantaged, they are more concerned about food on their table than what the tax code is, or who the Commissioner of Agriculture is, or who the Defense Secretary is. ", "They just want food in their stomach, clothes for their kids, and the ability to make a living, politics be damned.", "\n\nBut the flipside of losing interest in political processes or completely ignoring careers in public service irrespective of how less lucrative it is, is exemplified by cases like the Choke Hold in Staten Island, New York where an obvious strangulation of a Black man went unpunished.", "\n\nChances are that the Eric Garner case would have had its day in court if a Black man was the Richmond County District Attorney. ", "And rightfully so. ", "And even though the Feds by choosing not to charge Officer Darrell Wilson is substantiating the Saint Louis County prosecuting attorney’s decision not to prosecute, but if it was a Black man that handled that case, the outcome would have been no different, but the loss and destruction that followed the irresponsible handling of the grand jury decision would all have been avoided. ", "What am I saying? ", "I am not in any way implying that it is only when African American justice officers prosecute other African Americans that justice is served. ", "All I am saying here is that sometimes there are people in the justice system and in other aspects of the American public service sector who are not supposed to be there but who are regrettably there because folks who should’ve pursued a career in those public service areas and bring to the job the responsiveness and empathy that the job requires chose to steer clear of their civic duties.", "\n\nOur Founder with the Chief Prosecutor of the International Criminal Court, Mrs. Fatou Bensouda at the Kofi Annan International Peacekeeping Training Centre, Accra, Ghana. ", "May 20, 2011.", "\n\nI was having a conversation with couple of brothers not too long ago about the need for more African Americans to participate in the political process of this great nation and I was embarrassed by the response I got from them. ", "Some said that voting is a waste of time that it won’t change a thing. ", "Some said that seeking a career in public service to effect a change is a waste of time. ", "That the White man won’t allow you to advance to a position of relevance where you can actually make and implement policies that can effect a positive change in the society. ", "Some said that the White man just allowed Obama to be there to do their bidding.", "\n\nBut none of these are true. ", "First, if voting is a waste of time, why did people bleed and die to secure for us the right to vote? ", "Voting counts and voting matters. ", "The truth of the matter is that when we sit out elections, be it city, state or federal elections, we are regarding Dr. Martin Luther King Jr. and others like him as fools who died so we can have the right to vote. ", "And unintentionally, we are saying that the cause for which they laid down their lives was a wasted cause. ", "And that is not supposed to be so.", "\n\nAccording to Congressman John Lewis, a civil rights veteran, “The vote is precious. ", "It is almost sacred. ", "It’s the most powerful non-violent tool we have in a democratic society and we’ve got to use it.” ", "It comes handily and cost nothing comparable to protesting perceived injustice. ", "And for people who are still oppressed in the 21st century, there is no better tool for Black people to use to redress oppression than stepping out to vote, any and every time election calls. ", "Nobody should need to knock on your door to plead with you to go and exercise your right that was won for you at the altar of blood and sacrifice. ", "The Democratic Party need not raise money for get-out-the-vote drive so that they can come and prod and remind you to go exercise those rights. ", "You owe it to your Black self, you owe it to Selma, you owe it to Memphis Tennessee where Dr. King was murdered, you owe it to Emmett Till, you owe it to the Black Wall Street, you owe it to Robert F. Kennedy, you owe it to goddamed present and posterity to prioritize and get out of your lazy-and-full-of-excuses-and-full-of-shit ass to go and vote! ", "Go vote.", "\n\nFor every African American that have dared to dream about a reputable career in the United States Government, they all got to where they wanted to be, as long as they qualified for the positions they were shooting for. ", "The system didn’t try to stop them. ", "The system helped them on.", "\n\nI read in Times how Condoleezza Rice, the first female National Security Adviser (NSA) to a U.S. President, and the second female Secretary Of State in U.S. history, when she was a child and visited Washington D.C with her father. ", "As they were walking past the White House, hand in hand, she told her dad than one day she will occupy an office in the White House. ", "And that she did as the NSA to President George Walker Bush. ", "Colin Powell rose to the rank of a General in the U.S. Army and became the Chairman of the Joint Chiefs. ", "He was the NSA to Ronald Reagan and the first African-American Secretary of State. ", "Eric Holder has been the U.S. Attorney General for the past six years and is likely to be succeeded by Loretta Lynch, a Black woman and the first of her kind, if confirmed by the Senate, to occupy that position. ", "This is just to mention a few. ", "And Barack Obama, the son of a Kenyan immigrant is the President and Commander-In-Chief of the Armed Forces of the United States of America. ", "My people, these things are not fairy tales. ", "They are realities. ", "They are happening in our time and they are possible for everyone who dare to dream. ", "And we actually need more African Americans to dream of becoming something of a big player in the political theater of this country because the more we insert and integrate ourselves into the system, the less likely it is possible for those bag eggs in government to make and implement policies that are discriminatory and racists. ", "Because they will have us to contend with. ", "If the hen had attended the scheduled meeting like other animals did on that fateful day in the animal kingdom, first of all, in deference to her, no animal in its right mind would’ve dared suggest that the hen and her chicks be killed to provide the meat for refreshment for their upcoming ceremony. ", "All the animals would’ve put their heads together and reason out a way out that will be fair to all. ", "But because the hen chose to watch television, paint her nails, or play PlayStation while others were out debating important matters, she and her kids are made scapegoats. ", "Democracy is a representative system of government, when you are not represented, you and your interests will be overlooked.", "\n\nIt is outright ignorance and stupidity to suggest that President Obama was installed by White people to do their bidding. ", "Considering all the character assassination, vitriol, and stiff opposition to his progressive agenda that this young man has endured. ", "Agenda that has the capacity to set America apart from the rest of the world in terms of economic development, but agenda which the opposition regardless, opposes for no other reason than to deny him any lasting legacy. ", "For someone to turn around to say that he was a puppet of White people? ", "That is damn preposterous.", "\n\nAnd despite needless opposition, the president is still working for all Americans with sound and smart policies and programs. ", "And just like most policies and programs of presidents past, the fruits are not usually reaped while they are in office, but mostly after they have left. ", "And in due time, African Americans as well as White people, Native Americans, Hispanics, Asians, and immigrant Africans will reap the fruits of his policies and programs.", "\n\nEncouraging steps have been made by Whites and Blacks alike to achieve a more perfect and equitable union. ", "But more work need to be done. ", "And the bulk of it lies on the shoulders of today’s African American youths. ", "Therefore I want to encourage my African American brethren to spend more time thinking of what they CAN do to combat racism than they spend simply talking and getting mad about it. ", "Get involved. ", "Educate yourselves. ", "Read the New York Times, Washington Post, Foreign Policy, the Huffington Post, Time Magazine, the Economist, Newsweek and others. ", "And don’t tell me it’s the White man’s version of news. ", "It ain’t. ", "All news have elements of glossing in it to satisfy the needs of corporate bosses of media houses. ", "But a smart man can read between the lines and draw conclusions for himself. ", "Get smart. ", "I also encourage y’all to watch Meet the Press, Face the Nation, C-Span, and This Week to keep abreast of what is happening in the polity that you are part of. ", "For those social science students, I encourage you to seek internships and careers at our elite policy-making think tanks which includes but are not limited to Cato Institute, Council on Foreign Relations, Carnegie Endowment for International Peace, Milken Institute, The Heritage Foundation, and the United States Institute of Peace. ", "Get involved. ", "Join the military. ", "Join the pool of Foreign Service personnel. ", "Prepare yourselves for district attorneys and other positions of relevance Just like Abe Lincoln’s time came, yours will come. ", "Run for commissioners of agriculture, land, and whatever in your respective states. ", "Volunteer for political parties. ", "Jon the FBI, the CIA, the NSA. ", "Insert and integrate yourselves into every fabric of American body polity.", "\n\nThere are tons of scholarship opportunities out there for college degrees. ", "Explore them.", "\n\nThe Bill and Melinda Gates Foundation have the College-Ready Education and the Post-Secondary Success programs whose goals are to ensure that graduates from high school are made ready to succeed in college and to ensure that all students who seek the opportunity for college degrees are able to complete a high-quality postsecondary education that leads to a sustaining career. ", "Take advantage of it. ", "We also have Startup: Education, the Mark Zuckerberg Foundation, whose mission is to take a startup approach to improve education for all students. ", "Take advantage of that opportunity too. ", "Workforce Solutions in Texas awards full scholarship to Texas residents. ", "I believe other states in the union have similar scholarship programs. ", "I was a beneficiary of the Texas Workforce Solutions scholarship. ", "It covered my tuition, paid for all the books that I needed, and also provided money for miscellaneous expenditure such as printing and photocopying. ", "If an immigrant like me can access those educational opportunities, I believe citizens of this great nation can also access them more easily. ", "Take advantage. ", "The rich made themselves rich through hard work, determination, and focus. ", "Don’t envy the rich their wealth. ", "Don’t ask the government to take from them and hand it to you. ", "Prepare yourselves like they did and then key into the system and you will be rich like they are.", "\n\nThe truth of the matter is that whether you want to do bad, be a mediocre, or do good for yourself, they all require work and investment of time. ", "The amount of time it takes to sell drugs by the street corner or walk the street with the hope of being picked up for sex, probably by a sick person, one can invest same amount of time to read and assimilate two to three chapters of a college textbook at the public library. ", "Beyoncé and Jay-Z are no different from any hustler from Houston’s 3rd Ward. ", "What set them apart is how they utilized their time and what they channeled their energy to.", "\n\nOnly if that young man disillusioned about his future can see himself in Tim Scott, the junior Senator from South Carolina, a Black man elected to the U.S. Senate on the Republican Party platform. ", "Only if that young girl that thinks that the only way she can make a living is by walking the street can draw inspiration from Congresswoman Mia Love, a Haitian-American, a Mormon from the State of Utah, and a Republican.", "\n\nFinally, it is imperative for our parents to educate their wards on the necessity and glory of serving their country. ", "And for those parents that endured the civil rights era, it is also vital that they focus on teaching their children of the need to take advantage of their rights to vote and be voted for so that they can effect the change we want to see in America, rather than talking and reminiscing about racism, racism, racism. ", "Move on!", "\n\nFolks trooped to the cinemas to watch Selma. ", "Watching Selma in itself and seething with anger or repulsion at the evil that was done does not constitute an honor or tribute to those that died for our right to vote. ", "Taking the rightful step of participating in the political process with our votes, with our education, and by putting ourselves out there to be voted for so that we can serve with our expertise, is the only thing that will make the dream of an African American being judged by the content of his character rather than the color of his skin a reality." ]

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[ "Those who track and follow developments in the Indian smartphone market know it’s a jungle out there, with wafer-thin margins, cutthroat rivalries, and a spec-obsessed user base that can get quite finicky. ", "91mobiles’ Great Indian Smartphone Survey 2019 is an attempt to get inside the mind of smartphone users and to figure out whether these folks are satisfied with their brand of smartphone, the issues they face on a daily basis, the features they covet and the brands they want to migrate to next. ", "With over 15,000 entries, this survey represents the true voice of the user. ", "We already gave you a sneak peek at some of the insights we gained during this massive exercise, and now it’s time for the full lowdown.", "\n\n\n\n\n\nTrends in smartphone ownership\n\nBreakup of smartphone brands people own currently\n\nLooking at the split of smartphone brands used by respondents currently, we see that Xiaomi leads with a share of 23.6 percent, followed by Samsung with a share of 21.4 percent. ", "This is indicative of the respective market share of these brands, and doesn’t really come as a surprise. ", "The real insights however, appeared when we compared this data with the set of numbers we collected during the survey we conducted last year. ", "By comparing the data, we were able to ascertain brands that have improved in popularity this year, and those that have fallen out of favour.", "\n\n\n\nSmartphone brands with the highest gain in ownership share (from 2018 to 2019)\n\nThe numbers speak for themselves and there are no two ways about this – Realme is the brand that has defined convention when it comes to increase in popularity. ", "From a measly 0.30 percent share in 2018, Realme now has a very impressive 10 percent representation among survey respondents, and that shows how quickly this brand has grown, and to what extent. ", "Catering to the spec obsession of users in the affordable segment with devices featuring loaded configurations, attractive designs and even more attractive price tags, the brand is riding the popularity wave.", "\n\nThen there’s Samsung, which might have lost its market leader crown to Xiaomi in India, but is far from being down and out. ", "The Korean giant has fought back hard, especially this year, marginally increasing its share from 19 percent last year to 21.4 percent in 2019 (based on survey data). ", "Taking on the Chinese insurgents with its new M series that includes smartphones high on value and style, Samsung is taking the fight back to its rivals. ", "It has also launched quite a few noteworthy devices in the mid range (with its A series) and made the right moves with its 2019 flagships in the S series and Note range as well.", "\n\nVivo is another brand that has shown increase in popularity. ", "This mid-range brand played on aspects like design, cameras and specs and has been very aggressive this year in terms of launching new devices, following the latest smartphone trends and pricing the handsets well. ", "The efforts seem to have paid off, with Vivo’s share among our respondents growing from 5 percent to 7 percent.", "\n\nSmartphone brands with the steepest decline in ownership share (from 2018 to 2019)\n\nAmong brands that dropped in popularity, Motorola, Honor, Nokia, Huawei and OPPO are the ones that draw attention. ", "The reasons aren’t hard to fathom. ", "Honor just hasn’t shown the same level of aggressiveness this year as compared to the past, and doesn’t have many compelling new smartphones to offer. ", "Its parent company Huawei getting hit by a US trade ban has further compounded issues, which could be the reason why Huawei is also seeing a drop. ", "On the other hand, Motorola, which was once a favourite among those looking for bang for the buck, seems to have lost the plot when it comes to pricing its devices. ", "In recent months, it has been trying to bounce back though, and has launched a few smartphones (One Vision and One Action) that seem reasonably good. ", "But whether these handsets can pull users towards Motorola in swarms and hordes remains to be seen. ", "Nokia and OPPO are seeing marginal drops as well, and for these brands, the reason could just be the fact that the competition has been much more aggressive.", "\n\n\n\nThe big surprise here is Xiaomi. ", "The numero uno brand seems to be dipping in popularity, with almost 28 percent respondents owning the company’s devices in 2018 falling to 23.6 percent in 2019. ", "While Xiaomi hasn’t swayed much from its original promise of offering phones that score high on value, it seems it has been a little slow in adopting the latest trends. ", "While the competition has been quick to launch phones with features like multiple rear cameras, in-display fingerprint scanners, pop-up selfie snappers and water drop notches, Xiaomi has been late to the party. ", "Furthermore, aggressive competition from brands like Realme hasn’t helped its cause either.", "\n\nBreakup of smartphone brands people plan to buy next\n\nWe asked survey respondents about the smartphone brands they’d want to move to whenever they upgrade their handsets next, and the insights we got don’t come across as very surprising. ", "OnePlus stands out as the brand that’s likely to take top position on the podium, with the data projecting a share of over 18 percent in the months to come. ", "Clearly, the brand’s strategy of offering flagship smartphones at relatively affordable prices, and its efforts in developing a strong community of fans and users, has paid off. ", "OnePlus is followed by Samsung and Xiaomi, as expected.", "\n\n\n\nRealme appears as a surprise contended at number 4 though, but we already know how quickly it has been growing and expanding its user base.", "\n\nHow ownership trends might change in the future\n\nWhen we compared the data on current ownership with the numbers on future ownership, we saw some pretty interesting trends. ", "The comparison shows that OnePlus is the brand likely to gain the most users in the future, with the growth pegged at over 12 percent.", "\n\nApple, which has always been an iconic, aspirational brands for most, is also showing a healthy growth of over 6 percent. ", "The other brands likely to see growth are Realme and Huawei. ", "Brands like Honor, Motorola, Samsung and Xiaomi however, might see negative growth.", "\n\n\n\nIn Xiaomi’s case specifically, it seems that prospective buyers are missing a successor to last year’s POCO F1. ", "As you might recall, Xiaomi launched a new sub brand last year in the form of POCO, with its debut offering, the POCO F1 challenging OnePlus’ claim to the flagship killer crown. ", "The POCO F1 however, is still the lone ranger in the sub brand’s portfolio, with no signs of a successor so far. ", "This is probably the reason why the percentage of folks wanting to buy a Xiaomi handset has dipped significantly from 28 percent last year to 17 percent this year. ", "That figure of 28 percent from last year includes 9 percent who wanted to buy a POCO phone, so the reason for the lower number this year seems clear as night and day.", "\n\nSmartphone features that users want\n\nWe asked users about the features they miss the most in their current handsets. ", "Based on the responds, were able to draw up a list of the most coveted features, and waterproofing tops the list. ", "Waterproofing of course, is a very handy feature to have, and provides peace of mind even if you’re a careful user and unlikely to spill liquids on your phone. ", "This feature is mostly available on premium flagship smartphones, and judging by our survey, is very much in demand with users in other segments as well, given then over 20 percent users voted in favour.", "\n\n\n\nQuick charge support is also in demand, with 19.5 percent users voting in favour. ", "The quick charge feature has now trickled down from flagships to mid-range and affordable phones, and hopefully, would get ubiquitous very soon. ", "Next, over 16 percent users voted in favour of pop-up selfie cameras, so clearly, this is another feature users would like to see in their phones.", "\n\nDemographic trends in brand preferences and ownership\n\nHow smartphone ownership stacks up within different age groups\n\nAs part of the survey exercise, we were also able to draw up insights on smartphone brands split across users in various age groups. ", "In general, Xiaomi and Realme have a larger share among users in younger age groups. ", "Over 33 percent of respondents below the age of 22 own Xiaomi handsets, while the same figure for Realme stands at almost 17 percent.", "\n\n\n\nOn the other hand, Samsung and OnePlus are more popular among people in older age groups, with 31.6 percent users in the age group of 45 and above owning Samsung, and over 8 percent owning OnePlus phones.", "\n\nHow smartphone ownership compares between genders\n\nWe also split ownership data across genders to see how brand preferences are divided between men and women. ", "Among women, the two brands that definitely seem more in favour are OPPO and Vivo. ", "For OPPO, the ownership share stands at 11 percent for women, versus 6.2 percent men. ", "For Vivo, the figures are pegged at almost 12 percent women, and 8 percent men.", "\n\n\n\nMen it seems, prefer ASUS, OnePlus and Realme. ", "For ASUS, the ownership share is tilted heavily in favour of male users, with the numbers standing at 4 percent versus 2 percent of women. ", "For OnePlus, the same numbers stand at 7 percent men against 4 percent women. ", "And for Realme, it’s 12.3 percent men versus 9 percent women.", "\n\nUser satisfaction\n\nOverall user satisfaction with current smartphone\n\n\n\nUser satisfaction is of utmost importance for any brand, regardless of the product or service they might be selling. ", "This metric is even more critical for smartphone companies, just because of the competition that exists in this industry and the vast array of options available across different price segments. ", "It’s OnePlus that takes the crown, with almost 8 in 10 of the brand’s users saying they’re satisfied with their smartphone. ", "OnePlus is followed by Realme, with 65 percent users saying they’re satisfied with the handset, while Apple, with over 63 percent users voting in favour, stands at the third position.", "\n\n\n\nWe also asked users regarding satisfaction levels with specific aspects of their current phone, and derived data points to highlight. ", "Talking about satisfaction with battery for instance, it’s ASUS that has been rated at the top, with over 75 percent users saying they’re satisfied with the battery life offered by their ASUS handset. ", "ASUS is followed by OnePlus (70 percent), and Realme (67 percent).", "\n\n\n\nComing to camera capabilities, it’s Apple that has the highest number of satisfied users (72 percent), followed by OnePlus (70 percent) and Realme (63.5 percent). ", "OPPO also has a high number of users satisfied with the camera prowess of their phones, with almost 62 percent users voting in the affirmative.", "\n\n\n\nAs far as pure performance is concerned, it’s OnePlus that ranks at the top yet again, with over 88 percent users saying they’re satisfied with the performance. ", "Then there’s Apple at almost 80 percent, and Realme at 69 percent.", "\n\nComing to display capabilities, OnePlus occupies the top spot with almost 83 percent of its users saying they’re happy with their screens. ", "Apple ranks second with over 75 percent satisfied users, and Samsung follows at third spot with 71 percent.", "\n\nSoftware is a crucial aspect for a smartphone, but many tend to ignore it and look at only hardware specs instead. ", "In terms of user satisfaction based on software, thanks to 86 percent of its users voting in favour, OnePlus comes up at the top spot again. ", "Then there’s Apple with 81 percent votes, and Nokia with 67.5 percent satisfied users. ", "Surprisingly, Motorola ranks at the bottom of the list, which is strange since just like Nokia, most of Motorola’s phones run the stock Android One platform.", "\n\nService quality\n\nA whopping 55 percent survey respondents said they had to take their phone in for service at a service centre, which is a pretty high number if you ask us. ", "Clearly, smartphone brands need to do a lot more when it comes to quality control. ", "On a positive note, a large percentage of those who visited service centres rated their experience as very good or good.", "\n\n\n\nWhen the data on service quality was split by brand, it was OnePlus that took the lead with almost 79 percent users saying they are satisfied with the quality of service they received. ", "Apple follows with 78 percent votes, while OPPO appears as the best mid-range brand in the list with 64 percent votes.", "\n\nBrand loyalty among smartphone users\n\nThe most value for money brands\n\n\n\nOnePlus and Realme, unsurprisingly, rank at the top as smartphone brands delivering great value. ", "With almost 88 percent OnePlus users rating their phones 4 or higher (on a scale of 5) on value, the brand tops the chart. ", "Focussing on user experience, and offering solid specs at reasonable pricing has clearly worked in its favour. ", "Next up is Realme with 83.3 percent users rating it 4 or higher on value. ", "Its high-specced, yet affordable phones are the reason behind this.", "\n\nBrand loyalty\n\n\n\nA loyal set of users are what most brands want, since these folks not only come back as repeat customers, but also spread the word and recommend the brand to others. ", "In this all-important aspect, OnePlus leads yet again, with almost 63 percent of the brand’s users saying they’ll buy a device from the same company whenever they upgrade their phones next. ", "Apple follows closely with 62 percent votes. ", "Samsung leads as the brand that covers all price segments with over 44 percent votes.", "\n\nBrands with the highest jump in customer retention\n\n\n\nIn 2019, a larger number of Huawei and Apple users said they plan to buy from same brand when they upgrade next, compared to the same figures from last year. ", "In Huawei’s case, last year over 18 percent of its users had said they’d buy from the same brand, whereas this year, that number has jumped to 32 percent. ", "For Apple, the figure stood at 51 percent in 2018, but this year, 62 percent iPhone users have said they’d buy an Apple phone again, whenever they upgrade their daily drivers next.", "\n\nBrands with the highest fall in customer retention\n\n\n\nThings don’t seem to be working out too well for Honor, as last year, over 34 percent of its users had shown interest in buying a smartphone from the same brand again. ", "That number has dipped below 20 percent this year. ", "Similarly, in Nokia’s case, interest among its users in buying a Nokia phone again has dipped from 45 percent in 2018 to 31 percent this year.", "\n\nPossibly due to lack of a suitable successor in Xiaomi’s POCO range, and in part due to stronger competition, this Chinese giant seems to facing a dip too. ", "54 percent Xiaomi users wanted to buy their next smartphone from the same brand in 2018, but that number now stands at a little below 38 percent. ", "Maybe Xiaomi needs to reconsider its strategy with POCO and make sure its offerings tick most boxes in terms of features and design.", "\n\nSo there you have it. ", "Those are all the insights from our Great Indian Smartphone Survey 2019. ", "For a full copy of the report, please write in to us at survey@91mobiles.com." ]

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[ "Name\n\n ANGLE_platform_angle_device_type_swiftshader\n\nName Strings\n\n EGL_ANGLE_platform_angle_device_type_swiftshader\n\nContributors\n\n Jamie Madill, Google\n\nContacts\n\n Jamie Madill, Google (jmadill 'at' google 'dot' com)\n\nStatus\n\n Draft\n\nVersion\n\n Version 1, 2019-08-24\n\nNumber\n\n EGL Extension XXX\n\nExtension Type\n\n EGL client extension\n\nDependencies\n\n Requires EGL_ANGLE_platform_angle_vulkan.", "\n\nOverview\n\n This extension enables choosing the SwiftShader Vulkan implementation\n when it is available.", "\n\nNew Types\n\n None\n\nNew Procedures and Functions\n\n None\n\nNew Tokens\n\n Accepted as values for the EGL_PLATFORM_ANGLE_TYPE_ANGLE attribute:\n\n EGL_PLATFORM_ANGLE_DEVICE_TYPE_SWIFTSHADER_ANGLE 0x3487\n\nAdditions to the EGL Specification\n\n None.", "\n\nNew Behavior\n\n When calling eglGetPlatformDisplay:\n\n To request a SwiftShader-backed Vulkan implementation with ANGLE, the value\n of EGL_PLATFORM_ANGLE_TYPE_ANGLE should be EGL_PLATFORM_ANGLE_TYPE_VULKAN_ANGLE\n and the value of EGL_PLATFORM_ANGLE_DEVICE_TYPE_ANGLE should be\n EGL_PLATFORM_ANGLE_DEVICE_TYPE_SWIFTSHADER_ANGLE.", "\n\n If EGL_PLATFORM_ANGLE_DEVICE_TYPE_ANGLE is\n EGL_PLATFORM_ANGLE_DEVICE_TYPE_SWIFTSHADER_ANGLE and\n EGL_PLATFORM_ANGLE_TYPE_ANGLE is not\n EGL_PLATFORM_ANGLE_TYPE_VULKAN_ANGLE then an EGL_BAD_ATTRIBUTE\n error is generated and EGL_NO_DISPLAY is returned.", "\n\nIssues\n\n None\n\nRevision History\n\n Version 1, 2019-08-29 (Jamie Madill)\n - Initial draft\n" ]

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[ "\n784 F.Supp.2d 1102 (2011)\nSOUTHEAST ARKANSAS HOSPICE, INC., ", "Plaintiff\nv.\nDepartment of Health and Human Services, Kathleen SEBELIUS, Secretary, Defendant.", "\nCase No. ", "4:10CV00721 BSM.", "\nUnited States District Court, E.D. Arkansas, Western Division.", "\nMarch 24, 2011.", "\n*1104 Don E. Trimble, Attorney at Law, Little Rock, AR, for Plaintiff.", "\nShannon S. Smith, U.S. Attorney's Office, Little Rock, AR, for Defendant.", "\n\nORDER\nBRIAN S. MILLER, District Judge.", "\nPlaintiff Southeast Arkansas Hospice, Inc., (SEARK) filed this case against defendant Kathleen Sebelius, the Secretary of the United States Department of Health and Human Services (DHHS), seeking to invalidate the Secretary's regulation implementing a statutory cap on Medicare payments to hospice-care providers. ", "Four motions are pending: (1) the Secretary's motion to dismiss for lack of subject matter jurisdiction [Doc. ", "No. ", "17]; (2) the Secretary's motion to dissolve the preliminary injunction [Doc. ", "No. ", "29]; (3) SEARK's motion to amend its complaint [Doc. ", "No. ", "36]; and (4) SEARK's motion for summary judgment [Doc. ", "No. ", "38]. ", "For the reasons set forth below, the Secretary's motion to dismiss is granted in part and denied in part; the Secretary's motion to dissolve the preliminary injunction is denied as moot; SEARK's motion to amend its complaint is denied; and SEARK's motion for summary judgment is granted in part and denied in part.", "\n\nI. FACTUAL BACKGROUND\nIt is undisputed that SEARK operates hospice-care facilities in Arkansas and receives reimbursement from Medicare for certain services it provides to terminally ill patients. ", "See 42 U.S.C. § 1395f(a)(7) (2006). ", "To ensure that hospice care payments do not exceed the costs of treatment in a conventional setting, there is a \"cap\" on the total amount paid in reimbursements to SEARK for all eligible patients in any given fiscal year. ", "Id. § 1395f(i)(2)(A); H.R.Rep. ", "No. ", "98-333, at 1 (1983), U.S. Code Cong. & ", "Admin.", "News 1983, p. 1043. ", "This cap is codified at 42 U.S.C. § 1395f(i)(2), and the attendant regulation, which is at the heart of this lawsuit, is found at 42 C.F.R. § 418.309(b)(1). ", "A provider's annual cap is calculated by multiplying (a) a per-patient amount defined by statute and indexed for inflation, times (b) the number of Medicare beneficiaries in the hospice program during that fiscal year. ", "42 U.S.C. § 1395f(i)(2)(A).", "\nAs is typical in the hospice-care industry, SEARK receives Medicare payments through a fiscal intermediary. ", "SEARK's fiscal intermediary is Palmetto GBA (\"Palmetto\"). ", "The fiscal intermediary is responsible for reviewing a provider's claims and generally pays the provider upon invoice. ", "When a provider receives reimbursements *1105 from Medicare that exceed its annual cap, the fiscal intermediary issues a Notice of Program Reimbursement (NPR), which is essentially a demand for the overage. ", "Palmetto served four such demands on SEARK for fiscal years 2004, 2005, 2007, and 2009.", "\nUnder 42 U.S.C. § 1395oo(a), a provider may challenge an NPR so long as the amount in controversy is at least $10,000 and it requests a hearing before the Provider Reimbursement Review Board (PRRB) within 180 days after receipt of the demand. ", "If the provider is dissatisfied with the PRRB's ruling, it may obtain judicial review by filing a civil action within 60 days of the PRRB's final determination. ", "Id. § 1395oo(f)(1).", "\n\nA. Fiscal Year Ending 2004\n\nOn June 25, 2007, Palmetto sent an NPR to SEARK demanding that SEARK refund overpayments in the amount of $137,113 for the period of November 1, 2003, through October 31, 2004. ", "SEARK did not appeal this NPR to the PRRB.", "\n\nB. Fiscal Year Ending 2005\n\nOn September 26, 2006, Palmetto sent an NPR to SEARK demanding that SEARK refund overpayments in the amount of $358,170 for the period of November 1, 2004, through October 31, 2005. ", "SEARK timely appealed this determination to the PRRB. ", "The PRRB responded with a letter directed to SEARK's counsel that set forth certain deadlines in the administrative appeal process. ", "When SEARK failed to perfect its appeal by timely completing the enumerated tasks, the PRRB dismissed its appeal. ", "SEARK did not file a civil action to review this ruling within 60 days as required by statute.", "\n\nC. Fiscal Year Ending 2007\n\nOn January 7, 2009, Palmetto sent an NPR to SEARK demanding that SEARK refund overpayments in the amount of $16,383 for the period of November 1, 2006, through October 31, 2007. ", "SEARK did not appeal this NPR to the PRRB.", "\n\nD. Fiscal Year Ending 2009\n\nOn June 8, 2010, Palmetto sent an NPR to SEARK demanding that SEARK refund overpayments in the amount of $89,368 for the period of November 1, 2008, through October 31, 2009. ", "Two weeks later, SEARK filed this lawsuit. ", "On August 17, 2010, SEARK's motion for a temporary restraining order was granted. [", "Doc. ", "No. ", "14]. ", "On September 7, 2010, the parties appeared for a hearing to determine whether a preliminary injunction was appropriate. ", "The Secretary argued that SEARK's failure to exhaust its administrative remedies as to fiscal years ending 2004, 2007, and 2009, and to timely commence suit as to fiscal year ending 2005, deprives this court of subject matter jurisdiction. ", "SEARK countered that several exceptions to the doctrine of exhaustion were applicable and further noted that it had in fact recently commenced a timely appeal with the PRRB as to fiscal year 2009. ", "A preliminary injunction was imposed on September 24, 2010, preserving the status quo so that a meaningful decision could eventually be rendered on the merits. [", "Doc. ", "No. ", "26].", "\nIndeed, SEARK's appeal of the 2009 NPR was docketed by the PRRB as case number 10-1323 on September 14, 2010, and on October 5, 2010, the PRRB granted expedited judicial review. ", "Centers for Medicare and Medicaid Services (CMS), however, on its own motion, intervened on October 18, 2010, notifying SEARK and the PRRB that it intended to review \"whether the Board [the PRRB] properly *1106 determined jurisdiction.\" ", "In a 13-page order dated December 6, 2010, CMS vacated the PRRB's determination that the amount in controversy exceeded $10,000 by defining amount-in-controversy to mean the difference between the cap as calculated by the regulation and a theoretical cap purportedly calculated according to the statute.", "\n\nII. ", "LEGAL STANDARDS\nThe gist of the Secretary's motion to dismiss is that SEARK's failure to exhaust its administrative remedies precludes this court from having subject matter jurisdiction over its claims. ", "Accordingly, her motion to dismiss, or in the alternative, for summary judgment, will be initially construed as motion to dismiss under Federal Rule of Civil Procedure 12(b)(1). ", "The standards governing motions for leave to amend a complaint, for summary judgment, and to dissolve a preliminary injunction are also set forth herein.", "\n\nA. Motion to Dismiss for Lack of Subject Matter Jurisdiction\n\nBecause federal question jurisdiction under 28 U.S.C. § 1331 does not lie in Medicare reimbursement cases, see Shalala v. Ill. Council on Long Term Care, 529 U.S. 1, 10-20, 120 S.Ct. ", "1084, 146 L.Ed.2d 1 (2000), section 1395oo provides the sole means for a provider to obtain judicial review of the Secretary's overpayment determination. ", "In St. Joseph's Hospital of Kansas City v. Heckler, the Eighth Circuit Court of Appeals held that \"a provider's right to seek further review [of a fiscal intermediary's decision] is strictly limited\" and the court held that the threshold requirements of section 1395oo(a) — including the 180-day filing deadline — are jurisdictional in nature. ", "786 F.2d 848, 849-50, 853 (8th Cir.1986).", "\nSt. Joseph's distinguished cases in which a provider challenges the PRRB's determination that its appeal failed to meet the threshold requirements of section 1395oo(a) from those in which the provider blatantly failed to follow the administrative scheme, finding the former cases ripe for judicial review and the latter unripe. ", "Id. at 851. ", "Essentially, the rule is when there is no legitimate dispute that the provider's appeal was untimely or failed to establish the requisite amount-in-controversy, the PRRB does not have jurisdiction to render a final decision. ", "Because the right to judicial review attaches solely to a \"final decision\" by the PRRB and not the fiscal intermediary's determination, a district court lacks subject matter jurisdiction to hear a dispute between a provider and the Secretary that was not initially heard by the PRRB. ", "Id. at 852.", "\n\nB. Motion for Leave to Amend Complaint\n\nFederal Rule of Civil Procedure 15(a)(2) provides that leave to amend should be freely granted when justice so requires. ", "The amended complaint, however, should cure the defects in the original complaint; leave need not be granted where the proposed amendments would be futile. ", "See Humphreys v. Roche Biomedical Laboratories, Inc., 990 F.2d 1078, 1082 (8th Cir.1993).", "\n\nC. Motion for Summary Judgment\n\nSummary judgment is proper if, after viewing the evidence in the light most favorable to the nonmoving party, no genuine issues of material fact exist and the moving party is entitled to judgment as a matter of law. ", "Nelson v. Corr. ", "Med. ", "Servs., ", "533 F.3d 958, 961 (8th Cir.2008). ", "The Secretary cannot survive the motion for summary judgment merely by pointing to disputed facts; the facts in dispute must be material to the outcome of the case. ", "Anderson v. Liberty Lobby, Inc., 477 U.S. *1107 242, 247-48, 106 S.Ct. ", "2505, 91 L.Ed.2d 202 (1986). ", "If the facts alleged by the Secretary, when viewed in the light most favorable to her case, would not allow a reasonable factfinder to find in her favor, then summary judgment should be granted in favor of SEARK. ", "Bloom v. Metro Heart Group of St. Louis, Inc., 440 F.3d 1025, 1029 (8th Cir.2006).", "\n\nD. Motion to Dissolve a Preliminary Injunction\n\nUnder Federal Rule of Civil Procedure 59(e), an order may be amended to correct manifest errors or law or fact. ", "See Hagerman v. Yukon Energy Corp., 839 F.2d 407, 414 (8th Cir.1988).", "\n\nIII. ", "DISCUSSION\nThis section contains four distinct rulings: (1) although SEARK's claims for fiscal years 2004, 2005, and 2007 are not properly before this court, subject matter jurisdiction does lie for its fiscal year 2009 claim; (2) SEARK's motion to amend must be denied because it is futile; (3) the Secretary's regulation is invalid, and she will be enjoined from enforcing it as to SEARK's Medicare reimbursements during fiscal year 2009; and (4) the Secretary's motion to dissolve the preliminary injunction is moot.", "\n\nA. Subject Matter Jurisdiction\n\nSubject matter jurisdiction does not lie herein for SEARK's claims for fiscal years 2004, 2005, and 2007. ", "This is the case because SEARK failed to timely appeal the 2004 and 2007 NPRs to the PRRB. ", "Although SEARK did timely appeal the 2005 NPR, it did not bring suit within 60 days of the PRRB's dismissal of its appeal as required by statute. ", "Accordingly, these claims must be dismissed.", "\nThe 2009 NPR, however, was timely appealed to the PRRB, and the PRRB commenced expedited judicial review. ", "The Secretary argues, however, that because the PRRB's determination of jurisdiction below was purportedly improper, this court similarly is without jurisdiction because there is no \"final decision\" to review. ", "The rule from St. Joseph's, supra, however, strongly counsels against her position: \"[A] decision rejecting a claim for failure to fulfill a threshold requirement must be a final decision for purposes of judicial review. ", "Otherwise, the PRRB [or the Secretary] could effectively preclude any judicial review of its [or her] decisions simply by denying jurisdiction of those claims that it [or she] deems to be non-meritorious.\" ", "786 F.2d at 851.", "\nMoreover, a recent decision from the Ninth Circuit Court of Appeals casts significant doubt on the merits of CMS's December 6, 2010, administrative decision. ", "In Los Angeles Haven Hospice v. Sebelius, the Secretary argued that in order have Article III standing to challenge a regulation, a hospice-care provider \"must prove that it suffered a `net' increase in its liability for overpayments from the operation of the hospice cap regulation in [a given fiscal year], over and above the amount it would have been required to pay for the same period under the hospice cap statute or a hypothetical regulation drawn in conformity with the statute.\" ", "638 F.3d 644, 656 (9th Cir.2011). ", "The Ninth Circuit strongly rejected this position by noting that \"such an alternative calculation under a nonexistent regulation would necessarily be hypothetical and speculative in nature\" and that \"so long as [a hospice-care provider] can point to some concrete harm logically produced by 42 C.F.R. § 418.309(b), it has standing to challenge the hospice cap regulation even though in a prior, current, or subsequent fiscal year it may also have enjoyed some offsetting benefits from the operation of the current regulation.\" ", "Id.\n*1108 Another issue, not explicitly raised by the Secretary, is whether SEARK may be permitted to file suit first and then exhaust its administrative remedies second. ", "This is obviously neither the prescribed method under section 1395oo nor is it accepted under general principles of law. ", "If, however, it were held that this procedural anachronism mandates dismissal, it would be without prejudice and SEARK would be free to refile because SEARK timely exhausted its administrative remedies. ", "Therefore, in the interests of judicial economy and common sense, it is held that SEARK's claim as to fiscal year 2009 was exhausted below and that there is subject matter jurisdiction herein.", "\n\nB. SEARK's Motion for Leave to Amend\n\nSEARK requests leave to amend its complaint. ", "The proposed amendments include an additional claim for an NPR issued for fiscal year 2005, this one from its Helena location, and a statement that the PRRB recently rejected its belated appeals — made in September 2010 — for fiscal years ending 2004, 2005 (Pine Bluff location), and 2007. ", "Furthermore, the amended complaint requests application of equitable tolling to any claim barred by section 1395oo.", "\nLeave to amend need not be granted if the proposed amendments will not affect the outcome. ", "Here, SEARK's proposed amended complaint appears to do two things to invoke subject matter jurisdiction for fiscal years ending 2004, 2005, and 2007: (1) obtain a \"final decision\" from the PRRB denying its appeals for those years; and (2) argue that equitable tolling cures any failure to exhaust administrative remedies.", "\nThe first approach is too little, too late. ", "As discussed above at length, 42 U.S.C. § 1395oo is the provider's only key to the courthouse door. ", "That statute plainly requires a provider to appeal an NPR to the PRRB within 180-days of receipt. ", "Doing this many years later in order to obtain a \"final decision\" from which to appeal is insufficient.", "\nThe second equity-based approach asks for an application of equitable tolling. ", "SEARK's argument here is that because other district courts have almost uniformly invalidated the Secretary's regulation, she should not be able to apply it even if SEARK failed to challenge its application through the administrative process. ", "SEARK has not demonstrated, however, that either the Secretary, Palmetto, or the PRRB in some way thwarted its ability to exhaust its administrative remedies. ", "Indeed, each NPR contained clear instructions regarding the a provider's right to appeal the intermediary's determination of overpayment. ", "It appears that SEARK simply failed to do so. ", "Because SEARK has not identified special circumstances or improper conduct, the application of equitable tolling to its claims for fiscal years 2004, 2005, and 2007 is not appropriate.", "\n\nC. SEARK's Motion for Summary Judgment\n\nSummary judgment is appropriate because the Secretary's regulation is invalid as a matter of law. ", "Indeed, although courts should normally defer to regulations promulgated by the Secretary, see Chevron, U.S.A., Inc. v. Natural Resources Defense Council, Inc., 467 U.S. 837, 104 S.Ct. ", "2778, 81 L.Ed.2d 694 (1984), many district courts around the country have invalidated the regulation at issue. ", "See, e.g., Lion Health Svcs., ", "Inc. v. Sebelius, 689 F.Supp.2d 849 (N.D.Tex. ", "Feb.22, 2010); Legacy Health Care, Inc. v. Sebelius, No. ", "1:09CV00149 TS (D. Utah June 15, 2010); Autumn Light Hospice v. Seblius, 2010 WL 988470 (W.D.Okla. ", "Mar. 12, 2010); Sojourn Care, Inc. v. Leavitt, No. ", "07CV00375 GKF-PJC (N.D.Okla. ", "Feb. 7, 2007). ", "Recently, the Fifth and Ninth Circuits have *1109 affirmed the first round of these decisions. ", "See Los Angeles Haven Hospice, Inc. v. Sebelius, 638 F.3d 644 (9th Cir.2011); Lion Health Svcs., ", "Inc. v. Sebelius, 635 F.3d 693 (5th Cir.2011).", "\nIn accord with this groundswell of authority, and for the reasons set forth in the litany of authorities cited above, the Secretary's regulation, 42 C.F.R. § 418.309(b), is hereby declared invalid because it is contrary to the clear intent of Congress. ", "Furthermore, the Secretary is hereby permanently enjoined from enforcing it against SEARK as to fiscal year 2009.", "\n\nD. The Secretary's Motion to Dissolve the Preliminary Injunction\n\nBecause SEARK's claims for fiscal years 2004, 2005, and 2007 have been dismissed, the preliminary injunction entered on September 24, 2010, is vacated as to those years. ", "The present grant of summary judgment in favor of SEARK as to fiscal year ending 2009, however, imposes a permanent injunction as to that year. ", "Accordingly, the Secretary's motion to dissolve the preliminary injunction is denied as moot.", "\nIT IS THEREFORE ORDERED THAT\n1. ", "Defendant's motion to dismiss [Doc. ", "No. ", "17] is granted in part and denied in part;\n2. ", "Plaintiff's motion for leave to amend [Doc. ", "No. ", "36] is denied;\n3. ", "Plaintiff's motion for summary judgment [Doc. ", "No. ", "38] is granted in part and denied in part; and\n4. ", "Defendant's motion to dissolve the preliminary injunction [Doc. ", "No. ", "29] is denied as moot.", "\nAn appropriate judgment will issue.", "\n" ]

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[ "Every April, taxpayers and BIR offices are busy para sa filing ng income tax returns. ", "Kung ikaw ay Filipino citizen living in the Philippines, receiving income from sources within or outside the Philippines, kailangan mong magfile ng income tax return….", "\n\nImage from http://osxdaily.com/2011/12/01/decorate-a-christmas-tree-with-ipad-iphone-ornaments/ The typical Filipino is a heavy gadget user, so every Pinoy Christmas shopper usually have one or two gadgets in their Christmas shopping list. ", "The challenge with using gadgets is that their technology is constantly changing, and with…\n\nYou will finally get to see the Pope! ", "Whether indoor or outdoor venues ang inyong dadaluhan, best be prepared para maximum ang participation ninyo sa events. ", "1. ", "Check the Pope’s itinerary/schedule as there may be last minute changes. ", "Monitor www.PusoNgPamilya.com…\n\nWe’ve worked our hardest and saved our best and now it’s time to invest in real estate. ", "Excited man tayong bumili na ng property, kailangang maging maingat pa rin tayo at mapanuri. ", "Image taken from http://www.zipmatch.com/real-estate-news-and-advice/tips-and-advice/buyers/5-ways-to-overcome-fears-when-buying-a-property-in-the-philippines Here are some tips…" ]

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[ "Moldova (disambiguation)\n\nMoldova, officially the Republic of Moldova, is a country in southeastern Europe.", "\n\nMoldova or Moldavia may also refer to:\n\nPlaces\n\nHistorical\n\n Moldavia, a geographic and historical region, and former principality in Eastern Europe (1346–1859)\n Moldavia (region of Romania), one of the four historical regions in contemporary Romania\n Moldavian Democratic Republic, a short-lived state (1917–1918)\n Moldavian ASSR, an autonomous Soviet republic within Ukrainian SSR (1924–1940)\n Moldavian SSR, a republic of the Soviet Union (1940–1941 / 1944–1991)\n\nOther places\n Moldova Nouă, a town in Caraș-Severin County, Romania\n Moldova, Estonia, a village in Lüganuse Parish, Ida-Viru County, Estonia\n Moldova (river), a river in eastern Romania\n\nShips\n RMS Moldavia, British steam ship, as HMS Moldavia sunk during World War I \n MV Anne Scan, a Slovenian-built cargo ship also known as Moldavia\n\nOther uses\n Moldova (newspaper), a Romanian newspaper in the 1930s\n Editura Moldova, a Moldovan publishing house; See List of Romanian-language publishers\n 2419 Moldavia, an asteroid discovered in 1974\n\nSee also\n Moldavians (disambiguation)\n Moldovan (disambiguation)\n Moldava (disambiguation)\n Moldau (disambiguation)" ]

{ "pile_set_name": "Wikipedia (en)" }

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[ "Use of intravascular ultrasound to guide endovascular repair of popliteal artery aneurysms.", "\nTo evaluate the outcomes of intravascular ultrasound (IVUS) directed endovascular exclusion of popliteal artery aneurysm (PAA) using stent grafts. ", "Clinical data of seven patients who underwent endovascular exclusion of PAA using IVUS guidance between 2009 and 2011 were retrospectively analyzed. ", "Outcome measures included graft patency, endoleak, and clinical symptoms. ", "A total of seven patients were treated with Viabahn stent grafts using IVUS guidance. ", "No patients were lost to follow-up (mean 12 ± 2 months). ", "Patients had an average of two vessel runoff at the end of the procedure. ", "All patients were symptom free with patent stents and no evidence of endoleak at follow-up. ", "Use of IVUS during endovascular treatment of PAA allows the physician to fully appreciate the anatomy of PAA, including location and extent of thrombus burden and the diameter and location of desired landing zones for appropriate sealing of the aneurysm proximally and distally. ", "IVUS-guided treatment is associated with excellent outcomes and does not increase procedural radiation and contrast load." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Conversion to everolimus dramatically improves the prognosis of de novo malignancies after liver transplantation for alcoholic liver disease.", "\nDe novo malignancies are a main cause for late death after liver transplantation (LT). ", "Everolimus (ERL) is an immunosuppressive agent with antitumoral properties. ", "The aim of the present retrospective study was to identify prognostic factors, including conversion to ERL, for patients presenting non-cutaneous de novo solid organ malignancy after LT for alcoholic cirrhosis. ", "The study population consisted of 83 patients (presenting 100 tumors, including 75% of upper aerodigestive tract cancers), among the 398 patients who underwent LT for alcoholic cirrhosis in our center. ", "After diagnosis, ERL was introduced in 38 patients and calcineurin-inhibitor was discontinued in 64.1% of them. ", "Tumor stage was a significant prognostic factor with a one-yr survival at 82.6% for early stages, 63.4% for intermediate stages (N+) and 27.4% for disseminated diseases (p < 0.001). ", "Associated relative risk factor was 2.202 (95% CI 1.044-4.644) for intermediate stages and 5.743 (95% CI 2.436-13.541) for metastatic stages. ", "One- and five-yr survival was 77.4% and 35.2% in ERL group vs. 47.2% and 19.4% in the non-ERL group, respectively (p = 0.003). ", "The relative risk factor for ERL was 0.447 (95%CI 0.257-0.778). ", "Our results strongly suggest that conversion to ERL improves the prognosis of de novo malignancies after LT for alcoholic cirrhosis. ", "Prospective studies are needed to confirm this benefit." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Currency Wars - Russia Buys 20.7 Tonnes Of Gold In December; Netherlands Refutes IMF Gold Data\n\nUPDATE: Since we published our blog this morning, the Dutch central bank has denied that it added to its gold reserves in December.", "\n\nDe Nederlandsche Bank, the Dutch central bank has denied reports in Reuters, Bloomberg and picked up by GoldCore, that the bank had increased its gold holdings for the first time in sixteen years. ", "IMF data had shown that the Dutch had increased their holdings to 622.08 tonnes.", "\n\n“De Nederlandsche Bank has not increased its gold holdings. ", "Several media reported this Tuesday that based on IMF figures, DNB’s gold stock increased in December 2014. ", "This is incorrect,” it said on its website.", "\n\nThe DNB’s correct and current gold holdings consist of 19.691 million troy ounces (612.5 tonnes), the tenth largest holder of the metal in the world, according to the World Gold Council’s January data.", "\n\nWe believe that it is only a matter of time before a European or other central bank begins to emulate China and Russia and starts accumulating gold. ", "Today's error may portend tomorrow's reality. ", "It is important to note that while Dutch central bank gold accumulation would have been a very significant development, Russia's steady and robust accumulation of gold is very important. ", "It came at a time when some analysts were suggesting and there was much chatter that Russia would sell gold reserves.", "\n\nGOLDCORE's BLOG THIS MORNING:\n\nDUTCH AND RUSSIAN CENTRAL BANK BUY 30 TONNES OF GOLD IN DECEMBER\n\nRussia and surprisingly the Netherlands were the largest central bank buyers in December - accumulating a significant 30.34 tonnes between them as currency wars intensify.", "\n\n\n\nDemand for gold as a diversification and monetary asset continues to be very robust and central banks remain net buyers of gold which should be supportive of prices.", "\n\n\n\n\n\nThe Netherlands, which has the ninth-biggest gold reserves, raised its bullion holdings for the first time in 16 years. ", "It added 9.61 tonnes to bring total gold reserves to 622.08 tonnes.", "\n\n\n\nRussia raised its gold reserves for a ninth straight month in December as the country continued its multi month gold buying spree, adding to the fifth-biggest gold holdings in the world, data from the IMF showed yesterday.", "\n\n\n\nRussia continues to dollar cost average into gold and increased its bullion holdings by another hefty 20.73 tonnes to 1,208.23 tonnes in December.", "\n\n\n\nThe December figure for Russia, who have the fifth largest reserves in the world, brings their officially stated reserves to 1208.23 tonnes. ", "If this trend were to continue their officially stated reserves would increase 20.6% this year.", "\n\n\n\n\n\nGiven that Russia perceives itself to be under financial and economic attack from the West, there is the possibility that they are accumulating more gold than they are declaring officially to the IMF.", "\n\nThis is what the People’s Bank of China (PBOC) has been doing in recent years and there is little reason why Russia may not adopt the Chinese practice of not being transparent in this regard.", "\n\nThe Chinese government have been surreptitiously accumulating vast quantities of the metal in recent years and there is no reason to believe this buying will end in the coming months as geopolitical and monetary risks intensify.", "\n\nWestern central banks seem to be balking at what will be seen as the disastrous policy of dumping the gold owned by their populations onto the market. ", "The Gold Anti Trust Action Committee (GATA) have documented how this was done in order to suppress gold prices, in a bid to support and maintain faith in the dollar as reserve currency.", "\n\nAlready there are strong movements across Europe to have sovereign gold stored domestically. ", "The German and Dutch central banks have recently reported the repatriation of large volumes of their gold being held by central banks of foreign nations.", "\n\nIt is worth bearing in mind that both these countries are on the record as having drawn up contingency plans in the event of the failure of the Euro.", "\n\nThe Netherlands added 9.61 tonnes to it's official holdings in December, on top of the 122 tonnes of gold they shipped home from New York in November. ", "This represents the first increase in their official reserves since 1998. ", "The Dutch central bank's holdings have been unchanged since late 2008.", "\n\nThis further undermines the notion that the gold repatriation was simply a \"routine measure to instill public confidence in the ability of the central bank to manage crises.\"", "\n\nIt would appear the Dutch central bank has greater concerns than public confidence and may be actively preparing for the fall out from the ECB's QE programme - a programme to which they were opposed - and or a default by Greece, Spain, Portugal or Italy.", "\n\nAmong the many factors that may have motivated the Dutch central bank to buy gold may have been a shot across the bows to the ECB to remind them that the Netherlands is equipped and prepared to revert to the guilder, should Mario Draghi in the ECB go too far in terms of QE and the debasement of the euro.", "\n\nIt may also signal that they are concerned as to whether they will be able to repatriate the rest of their gold reserves.", "\n\nThis is an important development as it is the first time to our knowledge that a western central bank has actually purchased gold in volume since before the launch of the Euro. ", "While the central banks of China, Russia and ex Soviet states have been acquiring the precious metal hand over fist since the dress rehearsal crisis of 2008, western central banks gold reserves have remain unchanged - officially any way.", "\n\nThe gold repatriation movement represents a turning of the tide with regards to attitudes towards central bank omnipotence in managing paper currencies.", "\n\nThe Dutch purchase is noteworthy and it will be important to keep an eye on their demand in the coming months to see if this was a once off or the start of a trend of the Dutch central bank and other western central banks buying gold.", "\n\nThe announcement will likely spur other central banks to take precautions and acquire gold.", "\n\nRichard Russell - the godfather of financial newsletter writers - has recently made a stunning assertion about the gold markets. ", "The 91 year-old, who lived through the great depression and fought in World War II, is very gentle and humane in his writing. ", "He is not given to bouts of sensationalism.", "\n\nIn his most recent Dow Theory Letters he suggests that physical gold may not be available to buy at anywhere near current prices within the next year.", "\n\n\"There is a giant secret stirring under today's market. ", "China, India, Russia and almost every central bank is buying physical gold. ", "I'm guessing that within another year, physical gold will be swept off the market.”", "\n\nWe have long contended that this would likely materialise given the scale of the current crisis and the very small size of the physical gold market globally. ", "The purchase of 30 metric tonnes of gold in one month is a lot of physical gold as there is only some 170,000 metric tonnes of above ground gold.", "\n\nHowever, in dollar, pound or euro terms it is tiny. ", "30 metric tonnes of gold is only worth some $1.24 billion or less than one day of ECB QE and a tiny fraction of the value of stock and bond markets today and indeed of global foreign exchange reserves.", "\n\nThe smart money will continue to follow the Russian central bank example of gradually accumulating gold and dollar, euro or pound cost averaging into an allocated and segregated physical gold position.", "\n\n\n\nComprehensive Guide to Currency Wars: Bye Bye Petrodollar, Buy, Buy Gold\n\nMARKET UPDATE\n\nToday’s AM fix was USD 1,279.00, EUR 1,132.96 and GBP 848.48 per ounce.", "\n\nYesterday’s AM fix was USD 1,282.75, EUR 1,141.54 and GBP 854.60 per ounce.", "\n\nGold fell $13.30 or 1.03% to $1,280.40 per ounce yesterday and silver slid $0.41 or 2.24% to $17.89 per ounce.", "\n\n\n\nSilver in US Dollars - 5 Years (Thomson Reuters)\n\nGold in Singapore for immediate delivery was nearly unchanged at $1,282.55 an ounce in the evening.", "\n\nSingapore, premiums have dropped to 70 cents to $1 an ounce, compared with $1.20 earlier this month. ", "In Hong Kong, premiums were at 50-70 cents an ounce, down from $1 two weeks ago.", "\n\nIn London, spot gold hit $1,281.46 an ounce in early morning trading. ", "The gold price is down with profit taking following the Greek elections and ahead of the Fed meeting starting today.", "\n\nEuropean finance ministers are meeting to consider how to prevent a Greek default. ", "It was communicated that they want to work with new leader Alexis Tsipras, as long as he relinquishes his demands for a debt writedown.", "\n\nFurther turmoil in markets, sluggish global growth, ECB QE and the risk of a new Eurozone debt crisis are all bullish for gold and silver’s outlook.", "\n\nGold bullion shipments from Hong Kong to China dropped 32 percent in 2014. ", "Chinese imports from Hong Kong were 750 metric tons last year down from 1,108.8 tons in 2013, data from the Hong Kong Census and Statistics Department showed.", "\n\nHong Kong gold export data to China gets less relevant by the month and a better benchmark for Chinese demand is now SGE withdrawals which are running at a healthy clip - both in 2014 - and so far in 2015\n\nChinese demand remains very robust as seen in the more than 130 tonnes of SGE withdrawals in the first two weeks of the year.", "\n\nThe World Gold Council said in April that its long-term Chinese demand outlook remains intact as store of wealth demand is expected to expand to at least 1,350 tons by 2017 amid rising wealth. ", "Mainland demand was a record 1,275.1 tons in 2013, the council said in November.", "\n\nChina’s central bank cut interest rates for the first time in two years in November and the government accelerated the approval of infrastructure projects to spur growth, fueling a 53 percent gain in the Shanghai Composite Index last year. ", "Gold may benefit if investors pull back from the world’s best-performing stock market in 2014, according to UBS.", "\n\nWhile the world’s second-largest economy expanded 7.4 percent last year, the slowest pace since 1990, the global flow of gold from West to East will probably last for as long as two decades, the China Gold Association (CGA) said in June.", "\n\nSilver for immediate delivery climbed 0.3 percent to $17.98 an ounce. ", "Platinum was unchanged at $1,254.28 an ounce while palladium retreated 0.5 percent to $786.75 an ounce.", "\n\nGet Breaking News and Updates Here" ]

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[ "---\nabstract: 'We study large groups of birational transformations $\\Bir(X)$, where $X$ is a variety of dimension at least $3$, defined over $\\C$ or a subfield of $\\C$. Two prominent cases are when $X$ is the projective space $\\p^n$, in which case $\\Bir(X)$ is the Cremona group of rank $n$, or when $X \\subset \\p^{n+1}$ is a smooth cubic hypersurface. ", "In both cases, and more generally when $X$ is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from $\\Bir(X)$ to $\\Z/2$. As a consequence we also obtain that the Cremona group of rank $n \\ge 3$ is not generated by linear and Jonquières elements.'", "\naddress:\n- 'Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland'\n- 'Institut de Mathématiques de Toulouse UMR 5219, Université de Toulouse, UPS F-31062 Toulouse Cedex 9, France'\n- 'Laboratoire angevin de recherche en mathématiques (LAREMA), CNRS, Université d’Angers, 49045 Angers cedex 1, France'\nauthor:\n- 'Jérémy Blanc, Stéphane Lamy & Susanna Zimmermann'\nbibliography:\n- 'biblio.bib'\ntitle: Quotients of higher dimensional Cremona groups\n---\n\nIntroduction\n============\n\nHigher rank Cremona groups\n--------------------------\n\nThe *Cremona group of rank $n$*, denoted by $\\Bir_\\k(\\p^n)$, or simply $\\Bir(\\p^n)$ when the ground field $\\k$ is implicit, is the group of birational transformations of the projective space.", "\n\nThe classical case is $n=2$, where the group is already quite complicated but is now well described, at least when $\\k$ is algebraically closed. ", "In this case the Noether-Castelnuovo Theorem [@Cas; @Carraminana] asserts that $\\Bir(\\p^2)$ is generated by $\\Aut(\\p^2)$ and a single quadratic transformation. ", "This fundamental result, together with the strong factorisation of birational maps between surfaces helps to have a good understanding of the group.", "\n\nThe dimension $n\\ge 3$ is more difficult, as we do not have any analogue of the Noether-Castelnuovo Theorem (see §\\[sec:Generators\\] for more details) and also no strong factorisation. ", "Here is an extract from the article “Cremona group” in the Encyclopedia of Mathematics, written by V. Iskovskikh in 1987 (who uses the notation $\\Cr(\\p^n_\\k)$ for the Cremona group):\n\n> [ *One of the most difficult problems in birational geometry is that of describing the structure of the group $\\Cr(\\p^3_\\k)$, which is no longer generated by the quadratic transformations. ", "Almost all literature on Cremona transformations of three-dimensional space is devoted to concrete examples of such transformations. ", "Finally, practically nothing is known about the structure of the Cremona group for spaces of dimension higher than $3$.*]{} [@Encyclo]\n\nThirty years later, there are still very few results about the group structure of $\\Bir(\\p^n)$ for $n\\ge 3$, even if there were exciting recent developments using a wide range of techniques. ", "After the pioneering work [@Demazure] on the algebraic subgroups of rank $n$ in $\\Bir(\\p^n)$, we should mention the description of their lattices via $p$-adic methods [@CantatXie], the study of the Jordan property [@ProShra], and the fact that Cremona groups of distinct ranks are non-isomorphic [@CantatMorphisms].", "\n\nFor $n=3$, there is also a classification of the connected algebraic subgroups [@Ume; @BFT], and partial classification of finite subgroups [@Pro2011; @Pro2012; @Pro2014]. ", "There are also numerous articles devoted to the study of particular classes of examples of elements in $\\Bir(\\p^n)$, especially for $n$ small (we do not attempt to start a list here, as it would always be very far from exhaustive).", "\n\nThe question of the non-simplicity of Cremona groups of higher rank was up to now left open. ", "Using modern tools such as the Minimal model programme and factorisation into Sarkisov links, we will be able in this text to give new insight on the structure of the Cremona groups $\\Bir(\\p^n)$ and of its quotients.", "\n\nNormal subgroups\n----------------\n\nThe question of the non-simplicity of $\\Bir(\\p^n)$ for each $n\\ge 2$ was also mentioned in the article of V. Iskovskikh in the Encyclopedia:\n\n> [*It is not known to date $(1987)$ whether the Cremona group is simple.[@Encyclo]*]{}\n\nThe question was in fact asked much earlier, and is explicitly mentioned in a book by F. Enriques in 1895:\n\n> [*Tuttavia altre questioni d’indole gruppale relative al gruppo Cremona nel piano $($ed a più forte ragione in $S_n$ $n>2)$ rimangono ancora insolute; ad esempio l’importante questione se il gruppo Cremona contenga alcun sottogruppo invariante $($questione alla quale sembra probabile si debba rispondere negativamente$)$.\\\n> [@Enriques p. 116][^1]*]{}\n\nThe feeling expressed by F. Enriques that the Cremona group should be simple was perhaps supported by the analogy with biregular automorphism groups of projective varieties, such as $\\Aut(\\p^n) = \\PGL_{n+1}(\\k)$. In fact in the trivial case of dimension $n = 1$, we have $\\Bir(\\p^1)=\\Aut(\\p^1)=\\PGL_2(\\k)$, which is indeed a simple group when the ground field $\\k$ is algebraically closed. ", "Pushing further the analogy with algebraic groups, it was proved by the first author that when considered as a topological group, the Cremona group $\\Bir(\\p^2)$ is simple, in the sense that any proper Zariski closed normal subgroup must be trivial [@Blanc2010]. ", "This result was recently extended to arbitrary dimension by the first and third authors [@BZ].", "\n\nThe non-simplicity of $\\Bir(\\p^2)$ as an abstract group was proven, over an algebraically closed field, by S. Cantat and the second author [@CantatLamy]. ", "The idea of proof was to apply small cancellation theory to an action of $\\Bir(\\p^2)$ on a hyperbolic space. ", "A first instance of roughly the same idea was [@Danilov], in the context of plane polynomial automorphisms (see also [@FurterLamy]). ", "The modern small cancellation machinery as developed in [@DGO] allowed A. Lonjou to prove the non simplicity of $\\Bir(\\p^2)$ over an arbitrary field, and the fact that every countable group is a subgroup of a quotient of $\\Bir(\\p^2)$ [@Lonjou].", "\n\nAnother source of normal subgroups for $\\Bir(\\p^2)$, of a very different nature, was discovered by the third author, when the ground field is $\\R$ [@Zimmermann]. ", "In contrast with the case of an algebraically closed field where the Cremona group of rank 2 is a perfect group, she proved that the abelianisation of $\\Bir_\\R(\\p^2)$ is an uncountable direct sum of $\\Z/2$. Here the main idea is to use an explicit presentation by generators and relations. ", "In fact a presentation of $\\Bir(\\p^2)$ over an arbitrary perfect field is available since [@IKT], but because they insist in staying inside the group $\\Bir(\\p^2)$, they obtain very long lists. ", "In contrast, if one accepts to consider birational maps between non-isomorphic varieties, the Sarkisov programme provides more tractable lists of generators. ", "Using this idea together with results of A.-S. Kaloghiros [@Kaloghiros], the existence of abelian quotients for $\\Bir(\\p^2)$ was extended to the case of many non-closed perfect fields by the second and third authors [@LZ].", "\n\nThe present paper is a further extension in this direction, this time in arbitrary dimension, and over any ground field $\\k$ which is a subfield of $\\C$. Our first result is the following:\n\n\\[TheoremBirPnnotsimple\\] For each subfield $\\k\\subseteq\\C$ and each $n\\ge 3$, there is a group homomorphism $$\\Bir_\\k(\\p^n)\\tto \\bigoplus_{I} \\Z/2$$ where the indexing set $I$ has the same cardinality as $\\k$, and such that the restriction to the subgroup of birational dilatations given locally by $$\\left\\{(x_1,\\dots,x_n)\\mapsrat \\left(x_1\\alpha(x_2,\\cdots,x_n),x_2,\\dots,x_n\\right)\\mid \\alpha\\in\\k(x_2,\\dots,x_{n})^*\\right\\}$$ is surjective. ", "In particular, the Cremona group $\\Bir_\\k(\\p^n)$ is not perfect and thus not simple.", "\n\nWe give below a few immediate comments, and a quick preview of the rest of the introduction where we will present several statements that generalise or complement Theorem \\[TheoremBirPnnotsimple\\] in different directions.", "\n\nFirst we emphasise that this result contrasts with the situation in dimension 2 (over $\\C$). ", "Indeed, as $\\Bir_\\C(\\p^2)$ is generated by the simple group $\\Aut(\\p^2)=\\PGL_3(\\C)$ and one quadratic map birationally conjugated to a linear map, every non-trivial quotient of $\\Bir_\\C(\\p^2)$ is non-abelian and uncountable.", "\n\nAnother intriguing point at first sight is the indexing set $I$. We shall be more precise later, but the reader should think of $I$ as a kind of moduli space for some varieties of dimension $n-2$. Indeed to construct the group homomorphism we will see $\\p^n$ as being birational to a $\\p^1$-bundle over $\\p^{n-1}$, and each factor $\\Z/2$ is related to the choice of a general hypersurface in $\\p^{n-1}$ of sufficiently high degree, up to some equivalence.", "\n\nThe next natural question is to understand the kernel of the group homomorphism. ", "As will soon become clear, it turns out that $\\Aut(\\p^n)=\\PGL_{n+1}(\\k)$ is contained in the kernel. ", "This implies that the normal subgroup generated by $\\Aut(\\p^n)$ and any finite subset of elements in $\\Bir_\\k(\\p^n)$ is proper. ", "Theorem \\[TheoremTame\\] below will be a stronger version of this fact.", "\n\nOne can also ask about the possibility to get a homomorphism to a free product of $\\Z/2$, instead of a direct sum. ", "We will see that is is indeed possible, and is related to the existence of many conic bundle models for $\\p^n$ which are not pairwise square birational. ", "See Theorems \\[TheoremBirMori\\] and \\[TheoremManyCB\\] below.", "\n\nFinally, one can ask about replacing $\\p^n$ by a nonrational variety. ", "In this direction, we will prove the following result about the group $\\Bir(X)$ of birational transformations of a conic bundle $X/B$.\n\n\\[TheoremBirXBnotsimple\\] Let $B\\subseteq \\p^m$ be a smooth projective complex variety, $P\\to \\p^m$ a decomposable $\\p^2$-bundle $($projectivisation of a decomposable rank $3$ vector bundle$)$ and $X\\subset P$ a smooth closed subvariety such that the projection to $\\p^m$ gives a conic bundle $\\eta\\colon X\\to B$. Then there exists a group homomorphism $$\\Bir(X)\\tto \\bigoplus_\\Z \\Z/2$$ which is surjective in restriction to $\\Bir(X/B)=\\{\\phi\\in \\Bir(X)\\mid \\eta\\circ\\phi=\\eta\\}$.\n\nMoreover, if there exists a subfield $\\k\\subseteq \\C$ over which $X,B$ and $\\eta$ are defined, the image of elements of $\\Bir(X/B)$ defined over $\\k$ is also infinite.", "\n\nTheorem \\[TheoremBirXBnotsimple\\] applies to any product $X = \\p^1 \\times B$, to smooth cubic hypersurfaces $X \\subseteq \\p^{n}$ (see Section \\[Sec:CubicVar\\] and in particular Corollary \\[cor:SmoothCubicBirXZ2\\] and Proposition \\[Prop:Cubicfreeproduct\\]), and to many other varieties of dimension $n \\ge 3$ which are very far from being rational (see for instance [@KollarCB Theorem 3] and [@AhmOka Theorem 1.1 and Corollary 1.2]). ", "Of course it also includes the case of $X = \\p^1 \\times \\p^{n-1}$ which is birational to $\\p^n$, but observe that Theorem \\[TheoremBirPnnotsimple\\] is slightly stronger in this case, since there the set indexing the direct sum has the same cardinality as the ground field, and also because we can give an explicit subgroup, easy to describe, whose image is surjective.", "\n\nGenerators {#sec:Generators}\n----------\n\nAs already mentioned, the Noether-Castelnuovo theorem provides simple generators of $\\Bir(\\p^2)$ when $\\k$ is algebraically closed. ", "Using Sarkisov links, there are also explicit (long) lists of generators of $\\Bir(\\p^2)$ for each field $\\k$ of characteristic zero or more generally for each perfect field $\\k$ [@Isk1991; @Isk1996]. ", "In dimension $n\\ge 3$, we do not have a complete list of all Sarkisov links and thus are far from having an explicit list of generators for $\\Bir(\\p^n)$. The lack of an analogue to the Noether-Castelnuovo Theorem for $\\Bir(\\p^n)$ and the question of finding good generators was already cited in the article of the Encyclopedia above, in [@HMcK Question 1.6], and also in the book of Enriques:\n\n> [*Questo teorema non è estendibile senz’altro allo $S_n$ dove $n>2$; resta quindi insoluta la questione capitale di assegnare le più sempilici trasformazioni generatrici dell’intiero gruppo Cremona in $S_n$ per $n>2$.*]{} [@Enriques p. 115][^2]\n\nA classical result, due to H. Hudson and I. Pan [@Hudson; @Pan], says that $\\Bir(\\p^n)$, for $n\\ge 3$, is not generated by $\\Aut(\\p^n)$ and finitely many elements, or more generally by any set of elements of $\\Bir(\\p^n)$ of bounded degree. ", "The reason is that one needs at least, for each irreducible variety $\\Gamma$ of dimension $n-2$, one birational map that contracts a hypersurface birational to $\\p^1\\times \\Gamma$. These contractions can be realised in $\\Bir(\\p^n)$ by *Jonquières elements*, i.e. elements that preserve a family of lines through a given point, which form a subgroup $$\\PGL_2(\\k(x_2, \\dots, x_n)) \\rtimes \\Bir(\\p^{n-1}) \\subseteq \\Bir(\\p^n).$$\n\nHence, it is natural to ask whether the group $\\Bir(\\p^n)$ is generated by $\\Aut(\\p^n)$ and by Jonquières elements (a question for instance asked in [@PanSimis]).", "\n\nWe answer this question by the negative, in the following stronger form:\n\n\\[TheoremTame\\] Let $\\k$ be a subfield of $\\C$, and $n \\ge 3$. Let $S$ be a set of elements in the Cremona group $\\Bir_\\k(\\p^n)$ that has cardinality smaller than the one of $\\k$ $($for example $S$ finite, or $S$ countable if $\\k$ is uncountable$)$, and let $G \\subseteq \\Bir_\\k(\\p^n)$ be the subgroup generated by $\\Aut_\\k(\\p^n)$, by all Jonquières elements and by $S$.\n\nThen, $G$ is contained in the kernel of a surjective group homomorphism $$\\Bir_\\k(\\p^n) \\tto \\Z/2.$$ In particular $G$ is a proper subgroup of $\\Bir_\\k(\\p^n)$, and the same is true for the normal subgroup generated by $G$.\n\nIt is interesting to make a parallel between this statement and the classical Tame Problem in the context of the affine Cremona group $\\Aut(\\A^n)$, or group of polynomial automorphisms. ", "This is one of the “challenging problems” on the affine spaces, described by H. Kraft in the Bourbaki seminar [@KraftBourbaki]. ", "Recall that the tame subgroup $\\Tame(\\A^n) \\subseteq \\Aut(\\A^n)$ is defined as the subgroup generated by affine automorphisms and by the subgroup of elementary automorphisms of the form $(x_1, \\dots, x_n) \\mapsto (ax_1 + P(x_2, \\dots, x_n), x_2, \\dots, x_n)$. This elementary subgroup is an analogue of the $\\PGL_2(\\k(x_1, \\dots, x_n))$ factor in the Jonquières group, and of course the affine group is $\\PGL_{n+1}(\\k) \\cap \\Aut(\\A^n)$. The Tame Problem asks whether the inclusion $\\Tame(\\A^n) \\subseteq \\Aut(\\A^n)$ is strict in dimension $n \\ge 3$. It was solved in dimension $3$ over a field of characteristic zero in [@ShestakovUmirbaev], and remains an open problem otherwise.", "\n\nOn the one hand, one could say that our Theorem \\[TheoremTame\\] is much stronger, since we consider the *normal* subgroup generated by these elements, and we allow some extra generators. ", "It is not known (even if not very likely) whether one can generate $\\Aut(\\A^3)$ with linear automorphisms, elementary automorphisms and one single automorphism, and not even whether the normal subgroup generated by these is the whole group $\\Aut(\\A^3)$ (this last statement, even without the extra automorphism, seems more plausible).", "\n\nOn the other hand, even in dimension 3 we should stress that Theorem \\[TheoremTame\\] does not recover a solution to the Tame Problem. ", "Indeed, it seems plausible that the whole group $\\Aut(\\A^n)$ lies in the kernel of the group homomorphism to $\\Z/2$ of Theorem \\[TheoremTame\\]. ", "In fact, every element of $\\Bir(\\p^n)$ that admits a decomposition into Sarkisov links that contract only rational varieties (or more generally varieties birational to $\\p^2 \\times B$ for some variety $B$ of dimension $n-3$) is in the kernel of all our group homomorphisms (all are given by the construction of Theorem \\[TheoremBirMori\\] below), and it seems natural to expect that elements of $\\Aut(\\A^n)$ are of this type, but we leave this as an open question. ", "In fact we are not aware of any element of $\\Aut(\\A^3)$ which has been proved to lie outside the group generated, in $\\Bir(\\p^3)$, by linear and Jonquières maps: see [@BlancHeden Proposition 6.8] for the case of the Nagata automorphism, which can be generalised to any other automorphism given by a $\\mathbb{G}_a$ action, as all algebraic subgroups of $\\Bir(\\p^3)$ isomorphic to $\\mathbb{G}_a$ are conjugate [@BFT2].", "\n\nOverwiew of the strategy\n------------------------\n\nBy the Minimal model programme, every variety $Z$ which is covered by rational curves is birational to a Mori fibre space, and every birational map between two Mori fibre spaces is a composition of simple birational maps, called *Sarkisov links* (see Definition \\[def:sarkisovLink\\] and Theorem \\[thm:sarkisov\\]). ", "We associate to such a variety $Z$ the groupoid $\\BirMori(Z)$ of all birational maps between Mori fibre spaces birational to $Z$. The main idea is that even if we are primarily interested in describing morphisms from the group $\\Bir(Z)$ to $\\Z/2$, it turns out to be easier to first define such a morphism on the larger groupoid $\\BirMori(Z)$, and then restrict to $\\Bir(Z)$.\n\nWe concentrate on some special Sarkisov links, called *Sarkisov links of conic bundles of type* (see Definitions \\[def:sarkisovLink\\] and \\[def:4TypesOfLinks\\]). ", "To each such link, we associate a marked conic bundle, which is a pair $(X/B,\\Gamma)$, where $X/B$ is a conic bundle (a terminal Mori fibre space with $\\dim B=\\dim X-1$) and $\\Gamma\\subset B$ is an irreducible hypersurface (see Definition \\[def:markedConic\\] and Lemma \\[lem:SarkiIIConic\\]). ", "We also define a natural equivalence relation between marked conic bundles (Definition \\[def:markedConic\\]).", "\n\nFor each variety $Z$, we denote by $\\CB(Z)$ the set of equivalence classes of conic bundles $X/B$ with $X$ birational to $Z$, and for each class of conic bundles $C\\in \\CB(Z)$ we denote by $\\MC(C)$ the set of equivalence classes of marked conic bundles $(X/B,\\Gamma)$, where $C$ is the class of $X/B$.\n\nThe Sarkisov programme is established in every dimension [@HMcK] and relations among them are described in [@Kaloghiros]. ", "Inspired by the latter, we define [*rank $r$ fibrations*]{} $X/B$ (see Definition \\[def:rankFibration\\]); rank $1$ fibrations are Mori fibre spaces and rank $2$ fibrations dominate Sarkisov links (see Lemma \\[lem:rank 2 and link\\]). ", "We prove that the relations among Sarkisov links are generated by [*elementary relations*]{} (Definition \\[def:elementary relation\\]), which we define as relations dominated by rank $3$ fibrations (see Theorem \\[thm:sarkisov\\]).", "\n\nWe associate to each of these Sarkisov links $\\chi$ an integer $\\covgon(\\chi)$ that measures the degree of irrationality of the base locus of $\\chi$ (see §\\[sec:gonality\\]). ", "The BAB conjecture, proven in [@BirkarA] and [@BirkarS], tells us that the set of weak Fano terminal varieties of dimension $n$ form a bounded family and the degree of their images by a (universal) multiple of the anticanonical system is bounded by a (universal) integer $d$ (see Proposition \\[pro:BAB\\]). ", "As a consequence, we show that any Sarkisov link $\\chi$ of conic bundles of type appearing in an elementary relation over a base of small dimension satisfies $\\covgon(\\chi)\\leq d$ (see Proposition \\[pro:boundOnGenus\\]). ", "This and the description of the elementary relations over a base of maximal dimension and including a Sarkisov link of conic bundles of type (Proposition \\[pro:X4X3X2X1=id\\]) allows us to prove the following statement in §\\[sec:proofBirMori\\].", "\n\n\\[TheoremBirMori\\] Let $n\\ge3$. There is an integer $d > 1$ depending only on $n$, such that for every conic bundle $X/B$, where $X$ is a terminal variety of dimension $n$, we have a groupoid homomorphism $$\\BirMori(X) \\to \\bigast_{C\\in \\CB(X)} \\left(\\bigoplus_{\\MC(C)}\\Z/2\\right)$$ that sends each Sarkisov link of conic bundles $\\chi$ of type with $\\covgon(\\chi) > \\max\\{ d, 8\\conngon(X) \\}$ onto the generator indexed by its associated marked conic bundle, and all other Sarkisov links and all automorphisms of Mori fibre spaces birational to $X$ onto zero.", "\n\nMoreover it restricts to group homomorphisms $$\\begin{aligned}\n\\Bir(X) \\to \\bigast_{C\\in \\CB(X)} \\left(\\bigoplus_{\\MC(C)}\\Z/2\\right)\n,&& \\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2.\\end{aligned}$$\n\nIn order to deduce Theorem \\[TheoremBirPnnotsimple\\], we study the image of the group homomorphism from $\\Bir(X)$ and $\\Bir(X/B)$ provided by Theorem \\[TheoremBirMori\\], for some conic bundle $X/B$. In particular, we must check that these restrictions are not the trivial morphism. ", "We give a criterion to compute the image in §\\[sec:criterion\\]. ", "We apply this criterion to show that the image is very large if the generic fibre of $X/B$ is $\\p^1$ (or equivalently if $X/B$ has a rational section, or is equivalent to $( \\p^1\\times B)/B$). ", "This is done in §\\[sec:trivialBundle\\] and allows us to prove Theorem \\[TheoremBirPnnotsimple\\]. ", "Then in §\\[sec:nontrivialBundle\\] we study the more delicate case where the generic fibre $X/B$ is not $\\p^1$ (or equivalently if $X/B$ has no rational section), and show that for each conic bundle $X/B$, the image of $\\Bir(X/B)$ by the group homomorphism of Theorem \\[TheoremBirMori\\] contains an infinite direct sum of $\\Z/2$ (Proposition \\[pro:largeimageAnyCB\\]). ", "This allows to prove Theorem \\[TheoremBirXBnotsimple\\].", "\n\nNon-equivalent conic bundle structures\n--------------------------------------\n\nComing back to the case of $\\p^n$, we study the free product structure appearing in Theorem \\[TheoremBirMori\\]. ", "We want to prove that the indexing set $\\CB(\\p^n)$ is large. ", "This is equivalent to the question of existence of many non-equivalent conic bundle structures on $\\p^n$. Using conic bundles over $\\p^2$ with discriminant an elliptic curve, we manage to produce such examples, and we get the following.", "\n\n\\[TheoremManyCB\\] Let $n\\ge3$ and let $\\k\\subseteq \\C$ be a subfield. ", "There is a surjective group homomorphism $$\\Bir_\\k(\\p^n) \\tto \\bigast_{J} \\Z/2,$$ where the indexing set $J$ has the same cardinality as $\\k$. In particular, every group generated by a set of involutions with cardinality smaller or equal than $\\lvert \\k\\rvert$ is a quotient of $\\Bir_\\k(\\p^n)$. Moreover, the group homomorphism that we construct admits a section, so $\\Bir_\\k(\\p^n)$ is a semi-direct product with one factor being a free product.", "\n\nA first consequence is Theorem \\[TheoremTame\\]. ", "Other complements are given in Section \\[sec:Complements\\].", "\n\nFirst we get the SQ-universality of $\\Bir_\\k(\\p^n)$, meaning that any countable group is a subgroup of a quotient of $\\Bir_\\k(\\p^n)$. But in fact, many natural subgroups are quotients of $\\Bir_\\k(\\p^n)$, with no need to passing to a subgroup: this includes dihedral and symmetric groups, linear groups, and the Cremona group of rank 2 (see §\\[sec:SQ\\]).", "\n\nAnother consequence of our results is that the group $\\Bir_\\k(\\p^n)$ is not hopfian if it is generated by involutions, for each subfield $\\k\\subseteq \\C$ and each $n\\ge 3$ (Corollary \\[Cor:Hopfian\\]). ", "This is another difference with the dimension $2$, as $\\Bir_\\C(\\p^2)$ is hopfian and generated by involutions (see §\\[sec:Hopfian\\]).", "\n\nAll our results hold over any field abstractly isomorphic to a subfield of $\\C$ (§\\[sec:field\\]). ", "This is the case of most field of characteristic zero that are encountered in algebraic geometry. ", "For instance, any field of rational functions of any algebraic variety defined over a subfield of $\\C$.\n\nAnother feature of the Cremona groups in higher dimension is that the group $\\Bir_\\C(\\p^n)$ is a free product of uncountably many distinct subgroups, amalgamated over the intersection of the subgroups, which is the same for any two subgroups. ", "This strong version of an amalgamated product (Theorem \\[thm:AmalgamatedProduct\\]) is again very different from $\\Bir_\\C(\\p^2)$ (which is not a non-trivial amalgam, as already explained) and generalises to other varieties as soon as they have two non-equivalent conic bundle structures. ", "Again this result can be generalised to subfields of $\\C$.\n\nTheorem \\[thm:AmalgamatedProduct\\] implies that $\\Bir(\\p^n)$ acts non-trivially on a tree. ", "More generally, for each conic bundle $X/B$, we provide a natural action of $\\Bir(X)$ on a graph constructed from rank $r$ fibrations birational to $X$ (see §\\[sec:graph\\]).", "\n\nAknowledgements {#aknowledgements .unnumbered}\n---------------\n\nWe thank Hamid Ahmadinezhad, Marcello Bernardara, Caucher Birkar, Christian Böhnig, Hans-Christian Graf von Bothmer, Serge Cantat, Ivan Cheltsov, Tom Ducat, Andrea Fanelli, Enrica Floris, Jean-Philippe Furter, Philipp Habegger, Anne-Sophie Kaloghiros, Vladimir Lazić, Zsolt Patakfalvi, Yuri Prokhorov, Miles Reid and Christian Urech for interesting discussions related to this project.", "\n\nPreliminaries {#sec:preliminaries}\n=============\n\nUnless explicitly stated otherwise, all ambient varieties are assumed to be projective, irreducible, reduced and defined over the field $\\C$ of complex numbers.", "\n\nThis restriction on the ground field comes from the fact that this is the setting of many references that we use, such as [@BCHM; @HMcK; @Kaloghiros; @KKL]. ", "It seems to be folklore that all the results in these papers are also valid over any algebraically closed field of characteristic zero, but we let the reader take full responsibility if he is willing to deduce that our results automatically hold over such a field. ", "However, in Sections \\[sec:surjectivity\\] and \\[sec:freeProduct\\], see also §\\[sec:field\\], we will show how to work over fields that can be embedded in $\\C$.\n\nGeneral references for this section are [@KM; @LazarsfeldI; @BCHM].", "\n\nDivisors and curves {#sec:divisors}\n-------------------\n\nLet $X$ be a normal variety, $\\Div(X)$ the group of Cartier divisors, and $\\Pic(X) = \\Div(X)/\\sim$ the Picard group of divisors modulo linear equivalence. ", "The *Néron-Severi* space $N^1(X) = \\Div(X) \\otimes \\R / \\equiv$ is the space of $\\R$-divisors modulo numerical equivalence. ", "This is a finite-dimensional vector space whose dimension $\\rho(X)$ is called the *Picard rank* of $X$. We denote $N_1(X)$ the dual space of 1-cycles with real coefficients modulo numerical equivalence. ", "We have a perfect pairing $N^1(X) \\times N_1(X) \\to \\R$ induced by intersection. ", "If we need to work with coefficients in $\\Q$ we will use notation such as $N^1(X)_\\Q := \\Div(X) \\otimes \\Q / \\equiv$ or $\\Pic(X)_\\Q := \\Pic(X) \\otimes \\Q$. We say that a Weil divisor $D$ on $X$ is *$\\Q$-Cartier* if $mD$ is Cartier for some integer $m >0$. The variety $X$ is *$\\Q$-factorial* if all Weil divisors on $X$ are $\\Q$-Cartier. ", "An element in $\\Div(X) \\otimes \\Q$ is called a $\\Q$-divisor.", "\n\nFirst we recall a few classical geometric notions attached to a $\\Q$-divisor $D$. Let $m$ be a sufficiently large and divisible integer. ", "$D$ is effective, denoted $D \\ge 0$, if all coefficients of $D$ are non negative, and $D$ is *movable* if the base locus of the linear system $\\lvert mD \\rvert$ has codimension at least 2. ", "$D$ is *big* if the map associated with the linear system $\\lvert mD \\rvert$ is birational. ", "Similarly, $D$ is *semiample* if $\\lvert mD \\rvert$ is base point free, and $D$ is *ample* if furthermore the associated map is an embedding. ", "Finally, $D$ is *nef* if for any curve $C$ we have $D \\cdot C \\ge 0$.\n\nNow we recall how the numerical counterparts of these notions define cones in $N^1(X)$. The effective cone $\\Eff(X) \\subseteq N^1(X)$ is the cone generated by effective divisors on $X$. Its closure $\\widebar{\\Eff}(X)$ is the cone of *pseudo-effective* classes. ", "Similarly we denote $\\NE(X) \\subseteq N_1(X)$ the cone of effective 1-cycles, and $\\widebar{\\NE}(X)$ its closure. ", "By Kleiman’s criterion, a divisor $D$ is ample if and only if $D \\cdot C >0$ for any 1-cycle $C \\in \\widebar{\\NE}(X)$. It follows that the cone $\\Ample(X)$ of ample classes if the interior of the closed cone $\\Nef(X) \\subseteq N^1(X)$ of nef classes. ", "Similarly, the interior of the pseudo-effective cone $\\widebar{\\Eff}(X)$ is the big cone $\\BigD(X)$: Indeed a class $D$ is big if and only if $D \\equiv A + E$ with $A$ ample and $E$ effective. ", "A class is semiample if it is the pull-back of an ample class by a morphism. ", "Finally the *movable* cone $\\widebar{\\Mov}(X)$ is the closure of the cone spanned by movable divisors, and we will denote by $\\IntMov(X)$ its interior.", "\n\nOne should keep in mind the following inclusions between all these cones: $$\\begin{tikzcd}[column sep=small, row sep=small]\n\\Ample(X) \\ar[r,phantom,\"\\subseteq\"] &\n\\text{Semiample}(X) \\ar[r,phantom,\"\\subseteq\"] & \n\\Nef(X) \\ar[r,phantom,\"\\subseteq\"] \\ar[d,phantom,\"\\verteq\"] &\n\\widebar{\\Mov}(X) \\ar[r,phantom,\"\\subseteq\"] & \n\\widebar{\\Eff}(X) \\ar[d,phantom,\"\\verteq\"]\\\\\n&& \\widebar{\\Ample}(X) && \\widebar{\\BigD}(X)\n\\end{tikzcd}$$ We say that a 1-cycle $C \\in \\widebar{\\NE}(X)$ is *extremal* if any equality $C = C_1 + C_2$ inside $\\widebar{\\NE}(X)$ implies that $C, C_1, C_2$ are proportional.", "\n\nMaps\n----\n\nLet $\\pi\\colon X \\to Y$ be a surjective morphism between normal varieties. ", "We shall also denote $X/Y$ such a situation. ", "The *relative Picard group* is the quotient $\\Pic(X/Y):=\\Pic(X)/\\pi^*\\Pic(Y)$.\n\nWe say that a curve $C \\subseteq X$ is *contracted* by $\\pi$ if $\\pi(C)$ is a point. ", "The subsets $\\NE(X/Y) \\subseteq N_1(X/Y) \\subseteq N_1(X) $ are respectively the cone and the subspace generated by curves contracted by $\\pi$. The relative Néron-Severi space $N^1(X/Y)$ is the quotient of $N^1(X)$ by the orthogonal of $N_1(X/Y)$. The dimension $\\rho(X/Y)$ of $N^1(X/Y)$, or equivalently $N_1(X/Y)$, is the *relative Picard rank* of $\\pi$. If $\\pi$ has connected fibres, then $\\rho(X/Y) = 0$ if and only if $\\pi$ is an isomorphism, because a bijective morphism between normal varieties is an isomorphism.", "\n\nWe denote by $\\Eff(X/Y)$, $\\Nef(X/Y)$, $\\Ample(X/Y)$, $\\BigD(X/Y)$, $\\widebar{\\Mov}(X/Y)$ the images of the corresponding cones of $N^1(X)$ in the quotient $N^1(X/Y)$. If $D \\in N^1(X)$ is a class that projects to an element in $\\Nef(X/Y)$, we says that $D$ is $\\pi$-nef. ", "Equivalently, $D$ is $\\pi$-nef if $D \\cdot C \\ge 0$ for any $C \\in \\NE(X/Y)$. Similarly, we define the notion of $\\pi$-ample, $\\pi$-big, $\\pi$-effective. ", "In particular a class $D$ is $\\pi$-ample if $D \\cdot C > 0$ for any $C \\in \\widebar{\\NE}(X/Y)$.\n\nGeometrically, a $\\Q$-divisor $D$ is $\\pi$-ample if the restriction of $D$ to each fibre is ample, and $D$ is *$\\pi$-big* if the restriction of $D$ to the generic fibre is big. ", "We have the following characterisation for this last notion:\n\n\\[lem:piBig\\] Let $\\pi\\colon X \\to Y$ be a surjective morphism between normal varieties. ", "A divisor $D$ on $X$ is $\\pi$-big if and only if we can write $D$ as a sum $$D = \\pi\\text{-ample} + \\text{effective}.$$\n\nLet $D$ be a divisor on $X$, and $F = \\pi^{-1}(p)$ a general fibre. ", "By [@Nakayama Corollary 5.17] we have $$\\begin{gathered}\nD \\text{ is } \\pi\\text{-big} \\iff D|_F \\text{ is } \\text{big} \n\\iff D|_F = \\text{ample} + \\text{effective} \\\\ \n\\iff D = \\pi\\text{-ample} + \\pi\\text{-effective}.\\end{gathered}$$ We conclude by using the equivalence $$\\pi\\text{-effective} \\iff \\pi\\text{-numerically trivial} + \\text{effective},$$ and absorbing the $\\pi$-numerically trivial divisor in the $\\pi$-ample factor to get the result.", "\n\nWhen the morphism $\\pi\\colon X \\to Y$ is birational, the *exceptional locus* $\\Ex(\\pi)$ is the set covered by all contracted curves. ", "Assume moreover that $\\rho(X/Y) = 1$, and that $X$ is $\\Q$-factorial. ", "Then we are in one of the following situations [@KM Prop 2.5]: either $\\Ex(\\pi)$ is a prime divisor, and we say that $\\pi$ is a *divisorial contraction*, or $\\Ex(\\pi)$ has codimension at least 2 in $X$, and we say that $\\pi$ is a *small contraction*. ", "In this case, $Y$ is not $\\Q$-factorial.", "\n\nIn the context of a birational morphism, we have the following classical lemma. ", "Observe that here a class is $\\pi$-effective if it can be represented by an effective divisor with no component of the support in the exceptional locus of $\\pi$.\n\n\\[lem:Negativity\\] Let $\\pi \\colon X \\to Y$ be a birational morphism from a smooth variety $X$, $E$ a divisor with support in the exceptional locus of $\\phi$, and assume that $$E \\equiv \\pi\\text{-nef} + \\pi\\text{-effective}.$$ Then $-E \\ge 0$.\n\nGiven three normal varieties $X,Y,W$ together with surjective morphisms $X/W$, $Y/W$, we say that $\\phi\\colon X\\rat Y$ is a *rational map over $W$* if we have a commutative diagram $$\\begin{tikzcd}[link]\nX \\ar[rd] \\ar[rr,dashed,\"\\phi\"] && Y\\ar[ld] \\\\\n& W\n\\end{tikzcd}$$\n\nNow let $\\phi\\colon X \\rat Y$ be a birational map. ", "Any Weil divisor $D$ on $X$ is sent to a well-defined cycle $\\phi(D)$ on $Y$, and by removing all components of codimension $\\ge 2$ we obtain a well-defined divisor $\\phi_* D$: one says that $\\phi$ induces a map in codimension 1. ", "If $\\codim \\phi(D) \\ge 2$ (and so $\\phi_* D = 0$), we say that $\\phi$ contracts the divisor $D$. A *birational contraction* is a birational map such that the inverse does not contract any divisor, or equivalently a birational map which is surjective in codimension $1$. A *pseudo-isomorphism* is a birational map which is an isomorphism in codimension $1$. Birational morphisms and pseudo-isomorphisms (and compositions of those) are examples of birational contractions.", "\n\nWe use a dashed arrow $\\rat$ to denote a rational (or birational) map, a plain arrow $\\to$ for a morphism, and a dotted arrow $\\ps$, or simply a dotted line $\\dottedline$, to indicate a pseudo-isomorphism.", "\n\nWe denote by $\\Bir(X)$ the group of birational transformations of $X$. Given a surjective morphism $\\eta\\colon X \\to B$, we denote by $\\Bir(X/B)$ the subgroup of $\\Bir(X)$ consisting of all birational transformations over $B$, i.e. $$\\Bir(X/B) := \\{\\phi\\in \\Bir(X)\\mid \\eta \\circ \\phi=\\eta\\} \\subseteq \\Bir(X).$$\n\nMori dream spaces and Cox sheaves {#sec:moriDream}\n---------------------------------\n\nWe shall use a relative version of the usual definition of Mori dream space (compare with [@KKL Definition 2.2]). ", "Before giving the definition we recall the following notions.", "\n\nLet $\\pi\\colon X\\to Y$ be a surjective morphism, and $\\Fl$ a sheaf on $X$. The *higher direct images* of $\\Fl$ are the sheaves $R^i\\pi_* \\Fl$, $i \\ge 0$, which are defined on each affine subset $U \\subset Y$ as $R^i\\pi_* \\Fl(U) = H^i(\\pi^{-1}(U),\\Fl)$.\n\nWe say that a normal variety $Y$ has *rational singularities* if for some (hence any) desingularisation $\\pi\\colon X \\to Y$, we have $R^i\\pi_* \\Ol_X = 0$ for all $i > 0$.\n\nRecall also that a variety is *rationally connected* if any two general points are contained in a rational curve (see [@Kollar_rational IV.3]).", "\n\n\\[def:moriDream\\] Let $\\eta\\colon X\\to B$ be a surjective morphism between normal varieties. ", "We say that $X/B$ is a *Mori dream space* if the following conditions hold:\n\n1. ", " \\[dream:Q\\_factorial\\] $X$ is $\\Q$-factorial, and both $X,B$ have rational singularities.", "\n\n2. ", " \\[dream:rat\\_connected\\] A general fibre of $\\eta$ is rationally connected and has rational singularities.", "\n\n3. ", " \\[dream:nef\\] $\\Nef(X/B)$ is the convex cone generated by finitely many semiample divisors;\n\n4. ", " \\[dream:mov\\] There exist finitely many pseudo-isomorphisms $f_i\\colon X\\ps X_i$ over $B$, such that each $X_i$ is a $\\Q$-factorial variety satisfying \\[dream:nef\\], and $$\\widebar{\\Mov}(X/B)=\\bigcup f_i^*(\\Nef(X_i/B)).$$\n\n\\[lem:N1=Pic\\] Let $\\eta\\colon X\\to B$ be a surjective morphism between normal varieties, and $F$ a general fibre. ", "Assume that $X$ and $B$ have rational singularities, and assume:\n\n1. ", " \\[N1=Pic:F rat connected\\] $F$ is rationally connected and has rational singularities.", "\n\nThen the following properties hold true:\n\n1. ", " \\[N1=Pic:H1 F\\] $H^i(F,\\Ol_F) = 0$ for all $i > 0$;\n\n2. ", " \\[N1=Pic:R\\] $\\eta_* \\Ol_X = \\Ol_B$ and $R^i\\eta_* \\Ol_X = 0$ for all $i > 0$;\n\n3. ", " \\[N1=Pic:H1 U\\] $H^1(\\eta^{-1}(U), \\Ol_{\\eta^{-1}(U)}) = 0$ for each affine open set $U \\subset B$;\n\n4. ", " \\[N1=Pic:Pic XB\\] $\\Pic(X/B)_\\Q = N^1(X/B)_\\Q$.\n\nCondition \\[N1=Pic:F rat connected\\] from Lemma \\[lem:N1=Pic\\] is our condition \\[dream:rat\\_connected\\]. ", "The lemma implies that we would obtain a more general definition replacing \\[dream:rat\\_connected\\] by Condition \\[N1=Pic:H1 U\\], which is the choice of [@BCHM], or by Condition \\[N1=Pic:Pic XB\\], which is a relative version of the choice made in [@KKL]. ", "However our more restrictive definition suits to our purpose and seems easier to check in practice.", "\n\n\\[N1=Pic:F rat connected\\] $\\Longrightarrow$ \\[N1=Pic:H1 F\\]. ", "Consider a resolution of singularities $\\pi\\colon \\hat F\\rightarrow F$. Since $F$ has rational singularities, we have $R^i\\pi_*\\Ol_{\\hat F}=0$ for $i>0$. Then [@Hartshorne III, Ex.8.1] implies that $H^i(\\hat F,\\Ol_{\\hat F})\\simeq H^i(F,\\pi_*\\Ol_{\\hat F})=H^i(F,\\Ol_F)$ for all $i\\geq0$. Finally $H^i(F,\\Ol_F)=H^i(\\hat{F},\\Ol_{\\hat{F}})=0$ by [@Kollar_rational IV.3.8].", "\n\n\\[N1=Pic:F rat connected\\] $\\Longrightarrow$ \\[N1=Pic:R\\]. ", "Since $X$ has rational singularities, without loss in generality we can replace $X$ by a desingularisation and assume $X$ smooth. ", "We just saw that $H^i(F,\\Ol_F) = 0$ for all $i > 0$, and since we assume that $B$ has rational singularities, the result follows from [@Kollar_1986 Theorem 7.1].", "\n\n\\[N1=Pic:R\\] $\\Longrightarrow$ \\[N1=Pic:H1 U\\]. ", "This is just the definition of $R^1\\eta_* \\Ol_X = 0$.\n\n\\[N1=Pic:R\\] $\\Longrightarrow$ \\[N1=Pic:Pic XB\\]. ", "Let $D \\in \\Div(X)_\\Q$ a divisor which is numerically trivial against the contracted curves. ", "We want to show that $D$ is trivial in $\\Pic(X/B)_\\Q$, that is, a multiple of $D$ is a pull-back. ", "This is exactly the content of [@KollarMori1992 12.1.4]. ", "Observe that here again we only need the vanishing assumption for $i = 1$.\n\nLet $\\eta\\colon X \\to B$ be a surjective morphism between normal varieties, and $L_1$, $\\dots$, $L_r$ some $\\Q$-divisors on $X$. We define the divisorial sheaf $R(X/B; L_1, \\dots, L_r)$ to be the sheaf of graded $\\Ol_B$-algebras defined on every open affine set $U \\subset B$ as $$R(X/B;L_1,\\dots,L_r)(U) = \\bigoplus_{(m_1, \\dots,m_r) \\in \\N^r} H^0(\\eta^{-1}(U)/U, m_1L_1 + \\dots + m_rL_r),$$ where for any $D \\in \\Pic(X)_\\Q$ $$\\begin{gathered}\nH^0(\\eta^{-1}(U)/U, D) = \\\\ \n\\left\\{ f \\in \\k(\\eta^{-1}(U))^* \\mid \\exists L \\in \\Pic_\\Q(U), \\div(f) + D + \\eta^* L \\ge 0 \\right\\}\\cup \\{0\\}.\\end{gathered}$$\n\nIf moreover $\\Eff(X/B) \\subseteq \\sum \\R_+ L_i$, which ensures that we would get the same algebras using a $\\Z^r$-grading instead of $\\N^r$, then we say that the sheaf is a *Cox sheaf*, and we denote $$\\Cox(X/B;L_1,\\dots,L_r) := R(X/B;L_1,\\dots,L_r).$$ We say that a divisorial sheaf $R(X/B;L_1,\\dots,L_r)$ is finitely generated if for every affine set $U$ the $\\N^r$-graded $\\Ol_B(U)$-algebra $R(X/B;L_1,\\dots,L_r)(U)$ is finitely generated.", "\n\nAs the following lemma shows, for Cox sheaves this property of finite generation is independent of the choice of the $L_i$, and therefore we shall usually omit the reference to such a choice and denote a Cox sheaf simply by $\\Cox(X/B)$.\n\n\\[lem:FiniteGeneration\\] Let $\\eta\\colon X \\to B$ be a surjective morphism between normal varieties, $L_1, \\dots, L_r \\in \\Pic(X)_\\Q$ such that $\\Eff(X/B) \\subseteq \\sum \\R^+ L_i$, and $\\Cox(X/B;L_1,\\dots,L_r)$ the associated Cox sheaf. ", "Let $L_1',\\dots,L_s'\\in \\Pic(X)_\\Q$. If $\\Cox(X/B;L_1,\\dots,L_r)$ is finitely generated, then the divisorial sheaf $R(X/B;L_1',\\dots,L_s')$ also is finitely generated. ", "In particular, the property of finite generation of a Cox sheaf of $X/B$ does not depend on the choice of the $L_i$.\n\nAs already observed we can use a $\\Z^r$-grading and write $$\\Cox(X;L_1, \\dots, L_r)(U) = \\bigoplus_{(n_1, \\dots, n_r) \\in \\Z^r} H^0(\\eta^{-1}(U)/U, n_1 L_1 + \\dots + n_r L_r)$$ Replacing the $L_i$ par $\\frac1n L_i$ for some sufficiently divisible $n$, which by [@ADHL_2015 I.1.2.2] does not affect the finite generation, we can assume that each $L_j'$ is of the form $L_j' = n_1 L_1 + \\dots + n_r L_r+\\eta^* V$ for some $n_i \\in \\Z$ and some $\\Q$-divisor $V$ on $B$. So $$R(X; L_1', \\dots, L_r')(U) = \\bigoplus_{(m_1, \\dots, m_r) \\in \\N^r} H^0(\\eta^{-1}(U)/U, m_1 L_1' + \\dots + m_r L_r')$$ is a Veronese subalgebra of $\\Cox(X;L_1, \\dots, L_r)(U)$, and is finitely generated again by [@ADHL_2015 I.1.2.2].", "\n\n\\[lem:Mds\\] Let $X/B$ be a surjective morphism between normal varieties, whose general fibres are rationally connected. ", "Assume that $X$ is $\\Q$-factorial, and that $X$, $B$ and the general fibres have rational singularities. ", "Then $X/B$ is a Mori dream space if and only if its Cox sheaf is finitely generated.", "\n\nThe proof is similar to the proofs in the non-relative setting of [@KKL Corollaries 4.4 and 5.7].", "\n\n\\[ex:Mds\\] Standard examples of Mori dream spaces in the non relative case (i.e. when $B$ is a point) are toric varieties and Fano varieties. ", "Both of these classes of varieties are special examples of log Fano varieties, which are Mori dream spaces by [@BCHM Corollary 1.3.2]. ", "If $F$ is a log Fano variety, and $B$ is any smooth variety, then $(F \\times B) / B$ is a basic example of relative Mori dream space.", "\n\nMinimal model programme {#sec:MMP}\n-----------------------\n\nLet $X$ be a normal variety, and $C \\in \\widebar{\\NE}(X)$ an extremal class. ", "We say that the *contraction* of $C$ exists (and in that case it is unique), if there exists a surjective morphism $\\pi\\colon X \\to Y$ with connected fibres to a normal variety $Y$, with $\\rho(X/Y) = 1$, and such that any curve contracted by $\\pi$ is numerically proportional to $C$. ", "If $\\pi$ is a small contraction, we say that the *log-flip* of $C$ exists (and again, in that case it is unique) is there exists $X \\ps X'$ a pseudo-isomorphism over $Y$ which is not an isomorphism, such that $X'$ is normal and $X' \\to Y$ is a small contraction that contracts curves proportional to a class $C'$. For each $D \\in N^1(X)$, if $D'$ is the image of $D$ under the pseudo-isomorphism, we have a sign change between $D \\cdot C$ and $D' \\cdot C'$. We say that $X \\ps X'$ is a *$D$-flip*, resp.", " a *$D$-flop*, resp.", " a *$D$-antiflip* when $D \\cdot C < 0$, resp.", " $D \\cdot C = 0$, resp.", " $D \\cdot C > 0$.\n\nIf $D$ is nef on $X$, we say that $X$ is a *$D$-minimal model*. ", "If there exists a contraction $X \\to Y$ with $\\rho(X/Y) = 1$, $\\dim Y < \\dim X$ and $-D$ relatively ample, we say that $X/Y$ is a *$D$-Mori fibre space*.", "\n\nA *step* in the $D$-Minimal Model Programme (or in the $D$-MMP for short) is the removal of an extremal class $C$ with $D \\cdot C < 0$, either via a divisorial contraction, or via a $D$-flip. ", "In this paper we will ensure the existence of each step in a $D$-MMP by working in one the following contexts. ", "Either $D = K_X + \\Delta$ will be an adjoint divisor with $\\Delta$ ample and we can apply the main result of [@BCHM], or we will assume that $X$ is a Mori dream space, and rely on Lemma \\[lem:anyMMP\\] below (which is the reason for the name). ", "By *running a $D$-MMP from $X$*, we mean performing a sequence of such steps, replacing each time $D$ by its image, until reaching one of the following two possible outputs: a $D$-minimal model or a $D$-Mori fibre space. ", "In particular, observe that for us the output of a $D$-MMP is always of the same dimension as the starting variety, and the whole process makes sense even for $D$ not pseudo-effective (in contrast with another possible convention which would be to define the output of a $D$-MMP as $\\Proj(\\bigoplus_{n} H^0(X,nD))$.\n\nWe will often work in a relative setting where all steps are maps over a base variety $B$, and we will indicate such a setting by saying that we run a $D$-MMP *over $B$*.", "\n\nWhen $D = K_X$ is the canonical divisor, we usually omit the mention of the divisor in the previous notations. ", "So for instance given a small contraction contracting the class of a curve $C$, we speak of the flip of $C$ only if $K_X\\cdot C < 0$, of the $D$-flip of $C$ if $D\\cdot C <0$, and of the log-flip of $C$ when we do not want to emphasise the sign of the intersection against any divisor.", "\n\n\\[lem:anyMMP\\] If $X/B$ is a Mori dream space, then for any class $D \\in N^1(X)$ one can run a $D$-MMP from $X$ over $B$, and there are only finitely many possible outputs for such MMP.", "\n\nSingularities {#sec:singularities}\n-------------\n\nWe have recalled the basic terminology of the MMP without assumption on singularities, but as usual in practice we will make some drastic restriction on the allowed singularities. ", "A first basic fact is that $\\Q$-factoriality is preserved under all operations of the MMP. ", "Precisely, assume that $X$ is a normal $\\Q$-factorial variety. ", "If $\\pi\\colon X \\to Y$ is a divisorial contraction or a Mori fibre space, then $Y$ is $\\Q$-factorial. ", "If $\\pi\\colon X \\to Y$ is a small contraction and $X \\ps X'$ is the associated log-flip, then $X'$ is $\\Q$-factorial, but $Y$ is not (see [@KM 3.36 & 3.37]).", "\n\nNow let $X$ be a normal variety such that the canonical divisor $K_X$ is $\\Q$-Cartier, and let $\\pi\\colon Z \\to X$ be a resolution of singularities, with exceptional divisors $E_1$, $\\dots$, $E_r$. We say that $X$ has *terminal singularities*, or that $X$ is terminal, if in the ramification formula $$K_Z = \\pi^* K_X + \\sum a_i E_i,$$ we have $a_i >0$ for each $i$. Similarly we say that $X$ has *Kawamata log terminal* (*klt* for short) singularities, or that $X$ is klt, if $a_i > -1$ for each $i$. Each coefficient $a_i$, which is often called the *discrepancy* of $E_i$, does not depend on a choice of resolution in the sense that it is an invariant of the geometric valuation associated to $E_i$. Let $\\Delta$ an effective $\\Q$-divisor on $X$. We call $(X,\\Delta)$ a klt pair if $K_X+\\Delta$ is $\\Q$-Cartier and if for a (and hence any) resolution of singularities $\\pi\\colon Z\\to X$ such that the divisor $(\\pi^{-1})_* \\Delta \\cup\\Ex(\\pi)$ has normal crossing support we have $$K_Z=\\pi^*(K_X+\\Delta)+\\sum a_iE_i$$ where $\\pi_*(\\sum a_iE_i) +\\Delta=0$ and $a_i>-1$ for all $i$. Observe that if $(X,\\Delta)$ is a klt pair, then for any $\\Delta \\ge \\Delta ' \\ge 0$ the pair $(X,\\Delta')$ also is klt. ", "In particular taking $\\Delta' = 0$ we get that $X$ is klt.", "\n\n\\[lem:oppositeSigns\\] Let $\\pi\\colon X \\to Y$ be a divisorial contraction with exceptional divisor $E = \\Ex(\\pi)$ that contracts the class of a curve $C$. ", "If $D \\in \\Div(X)$ and $D' = \\pi_* D$, then in the ramification formula $$D = \\pi^* D' + aE,$$ the numbers $a$ and $D \\cdot C$ have opposite signs. ", "In particular, if $X$ is $\\Q$-factorial and terminal, then $Y$ is $\\Q$-factorial and terminal if and only if $K_X \\cdot C <0$.\n\nWe have $D \\cdot C = a E \\cdot C$, so the claim follows from $E \\cdot C < 0$. For this, see for instance [@Matsuki proof of 8-2-1(i)]. ", "The last assertion follows by taking $D=K_X$ and $D'=K_Y$.\n\nIf we start with a $\\Q$-factorial terminal variety and we run the classical MMP (that is, relatively to the canonical divisor), then each step (divisorial contraction or flip) of the MMP keeps us in the category of $\\Q$-factorial terminal varieties (for divisorial contractions, this follows from Lemma \\[lem:oppositeSigns\\]). ", "Moreover, when one reaches a Mori fibre space $X/B$, the base $B$ is $\\Q$-factorial as mentioned above, but might not be terminal. ", "However by the following result $B$ has at worst klt singularities.", "\n\n\\[pro:sing\\_of\\_B\\] Let $X /B$ be a Mori fibre space, where $X$ is a $\\Q$-factorial klt variety. ", "Then $B$ also is a $\\Q$-factorial klt variety.", "\n\nWe will also use the following related result:\n\n\\[pro:sing\\_of\\_Proj\\] Let $(X,\\Delta)$ be a klt pair, and set $$Y = \\Proj \\Bigl( \\bigoplus_m H^0(X, m(K_X + \\Delta)) \\Bigr)$$ where the sum is over all positive integers $m$ such that $m(K_X + \\Delta)$ is Cartier. ", "Then $Y$ is klt.", "\n\nThe following class of Mori fibre spaces will be of special importance to us.", "\n\n\\[def:conicBundle\\] A *conic bundle* is a $\\Q$-factorial terminal Mori fibre space $X/B$ with $\\dim B = \\dim X - 1$. The *discriminant locus* of $X/B$ is defined as the union of irreducible hypersurfaces $\\Gamma\\subset B$ such that the preimage of a general point of $\\Gamma$ is not irreducible. ", "We emphasise that the terminology of conic bundle is often used in a broader sense (for instance, for any morphism whose general fibre is isomorphic to $\\P^1$, with no restriction on the singularities of $X$ or on the relative Picard rank), but for our purpose we will stick to the above more restricted definition.", "\n\nWe say that two conic bundles $X/B$ and $X'/B'$ are *equivalent* if there exists a commutative diagram $$\\begin{tikzcd}[link]\nX \\ar[dd,swap]\\ar[r,\"\\psi\",dashed]& X'\\ar[dd] \\\\ \\\\\nB \\ar[r,\"\\theta\",dashed]& B' \n\\end{tikzcd}$$ where $\\psi, \\theta$ are birational.", "\n\nThe singular locus of a terminal variety has codimension at least $3$ ([@KM 5.18]). ", "This fact is crucial in the following result.", "\n\n\\[lem:divContToCodim2\\] Let $\\pi\\colon X \\to Y$ be a divisorial contraction between $\\Q$-factorial terminal varieties, with exceptional divisor $E$, and assume that $\\Gamma = \\pi(E)$ has codimension $2$ in $Y$. Then $\\pi$ is the blow-up of the symbolic powers of the sheaf of ideals $\\Il$ defining the reduced scheme $\\Gamma$. In particular, the fibre $f$ over a general point of $\\Gamma$ $($precisely, a point that is smooth for both $\\Gamma$ and $Y)$ is a smooth rational curve such that $K_X \\cdot f = E \\cdot f = -1$.\n\nThe fact that $\\pi$ is the blow-up of the sheaf of ideals $\\Il$ follows from the universal property of blowing-up ([@Hartshorne II.7.14]) and the assumption $\\rho(X/Y) = 1$. For the last assertion, it suffices to notice that since $X$ and $Y$ are terminal, there exists a codimension 3 closed subset $S \\subset Y$ such that $\\Gamma \\setminus S$, $Y \\setminus S$ and $X \\setminus \\pi^{-1}(S)$ are smooth, so that the restriction of $\\pi$ to $X \\setminus \\pi^{-1} (S)$ is the ordinary blow-up of a smooth subvariety.", "\n\n\\[lem:fibresAreTerminal\\] Let $\\eta\\colon X \\to B$ be a morphism between normal varieties with $X$ terminal $($resp.", " klt$)$. Then for a general point $p \\in B$, the fibre $\\eta^{-1}(p)$ also is terminal $($resp.", " klt$)$, so in particular it has rational singularities.", "\n\nThe fact that $\\eta^{-1}(p)$ is terminal (resp.", " klt) follows from [@Kollar_SantaCruz 7.7] by taking successive hyperplane sections on $B$ locally defining $p$. As already mentioned klt singularities are rational, see [@KM 5.22].", "\n\n\\[lem:fibresRatConnected\\] \n\n1. ", " \\[connected:-KBigNef\\] Let $(X,\\Delta)$ be a klt pair, and $\\pi\\colon X \\to Y$ be a morphism with connected fibres such that $-(K_X+\\Delta)$ is $\\pi$-big and $\\pi$-nef. ", "Then for every $p \\in Y$ the fibre $\\pi^{-1}(p)$ is covered by rational curves, and for a general $p \\in Y$ the fibre $\\pi^{-1}(p)$ is rationally connected with klt singularities.", "\n\n2. ", " \\[connected:birational\\] Let $Y$ be a klt variety, and $\\pi\\colon X \\to Y$ a birational morphism. ", "Then every fibre of $\\pi$ is covered by rational curves.", "\n\n3. ", " \\[connected:flips\\] Let $\\phi\\colon X \\ps X'$ be a sequence of log-flips between klt varieties, and $\\Gamma \\subset X$ a codimension $2$ subvariety contained in the base locus of $\\phi$. Then $\\Gamma$ is covered by rational curves.", "\n\nWe first prove \\[connected:-KBigNef\\]. ", "By [@HMcK2007 Corollary 1.3(1)], every connected component of every fibre of $\\pi$ is rationally chain connected. ", "As $\\pi$ has connected fibres, every fibre $f$ is rationally chain connected. ", "This implies that $f$ is uniruled [@Kollar_rational IV.3.3.4] and thus that $f$ is covered by rational curves [@Kollar_rational IV.1.4.4]. ", "The fact that a general fibre $f$ is klt is Lemma \\[lem:fibresAreTerminal\\]. ", "As $f$ is rationally chain connected, it is also rationally connected by [@HMcK2007 Corollary 1.5(2)].", "\n\n\\[connected:birational\\]: By [@HMcK2007 Corollary 1.5(1)], each fibre $f$ of $Y\\to X$ is rationally chain connected. ", "By the same argument as before, this implies that $f$ is uniruled and then covered by rational curves.", "\n\nFinally we prove \\[connected:flips\\]. ", "It is sufficient to consider the case of a single log-flip $X \\ps X'$, associated to a small contraction $X \\to Y$, and to prove that the exceptional locus of $X/Y$ is covered by rational curves. ", "If $-K_X$ is relatively ample, this follows directly from \\[connected:-KBigNef\\], applied with $\\Delta = 0$. Otherwise, denoting $C$, resp. ", "$C'$, a curve contracted by $X/Y$, resp. ", "by $X'/Y$, we have $K_X \\cdot C \\ge 0$ and $K_{X'}\\cdot C' \\le 0$. We choose an effective ample divisor $\\Delta'$ on $X'$ such that $(K_{X'} + \\Delta')\\cdot C' > 0$ and $(X',\\Delta')$ is a klt pair. ", "Denote by $\\Delta$ the strict transform of $\\Delta'$ on $X$. Then $(X,\\Delta)$ is a klt pair and $(K_X + \\Delta)\\cdot C < 0$ because a log-flip changes the sign of intersection, and again we can apply \\[connected:-KBigNef\\].", "\n\n\\[lem:B’\\_over\\_B\\_is\\_Mds\\] Let $X/Y$ be a Mori dream space that factorises as a composition of two morphisms $X/W$ and $W/Y$ via a $\\Q$-factorial klt variety $W$. Then $W/Y$ is a Mori dream space.", "\n\nThe general fibres of $W/Y$ are rationally connected because they are images of the rationally connected fibres of $X/Y$, and they have rational singularities by Lemma \\[lem:fibresAreTerminal\\]. ", "For any affine open subset $U \\subset Y$, the algebra $\\Cox(W/Y)(U)$ embeds by pull-back as a subalgebra of $\\Cox(X/Y)(U)$, hence is finitely generated by Lemma \\[lem:FiniteGeneration\\]. ", "We conclude by Lemma \\[lem:Mds\\].", "\n\nTwo-rays game {#sec:2rays}\n-------------\n\nA reference for the notion of two-rays game is [@Corti_explicit §2.2]. ", "We use a slightly different setting in the discussion below. ", "Namely, first we ensure that all moves do exist by putting a Mori dream space assumption, and secondly we do not put strong restrictions on singularities (this will come later in Definition \\[def:rankFibration\\]).", "\n\nLet $Y \\to X$ be a surjective morphism between normal varieties, with $\\rho(Y/X) = 2$. Assume also that there exists a morphism $X/B$ such that $Y/B$ is a Mori dream space. ", "In particular, by Lemma \\[lem:anyMMP\\] for any divisor $D$ on $Y$ one can run a $D$-MMP over $B$, hence a fortiori over $X$. Then $\\NE(Y/X)$ is a closed 2-dimensional cone, generated by two extremal classes represented by curves $C_1, C_2$. Let $D = -A$ where $A$ is an ample divisor on $Y$, so that a $D$-minimal model does not exist. ", "Then by Lemma \\[lem:anyMMP\\] for each $i = 1,2$ we can run a $D$-MMP from $Y$ over $X$, which starts by the divisorial contraction or log-flip of the class $C_i$, and produce a commutative diagram that we call the *two-rays game* associated to $Y/X$ (and which does not depend on the choice of $D$): $$\\begin{tikzcd}[link]\nY_1 \\ar[dd] & Y \\ar[l,dotted] \\ar[r,dotted] \\ar[ddd] & Y_2 \\ar[dd] \\\\ \\\\\nX_1 \\ar[dr] && X_2 \\ar[dl] \\\\\n& X &\n\\end{tikzcd}$$ Here $Y \\ps Y_i$ is a (possibly empty) sequence of $D$-flips, and $Y_i \\to X_i$ is either a divisorial contraction or a $D$-Mori fibre space.", "\n\nNow we give a few direct consequences of the two-rays game construction.", "\n\n\\[lem:2raysOverFlip\\] Let $Y_1/B$ be a Mori dream space, $Y_1 \\to X_1$ a morphism over $B$ with $\\rho(Y_1/X_1) = 1$, and $X_1 \\ps X_2$ a sequence of relative log-flips over $B$. Then there exists a sequence of log-flips $Y_1 \\ps Y_2$ over $B$ such that the induced map $Y_2 \\to X_2$ is a morphism, of relative Picard rank $1$ by construction. ", "Moreover if $Y_1/X_1$ is a divisorial contraction $($resp.", " a Mori fibre space$)$, then $Y_2/X_2$ also is.", "\n\nBy induction, it is sufficient to consider the case where $X_1 \\ps X_2$ is a single log-flip over a non $\\Q$-factorial variety $X$ dominating $B$, given by a diagram $$\\begin{tikzcd}[link]\nX_1 \\ar[rd] \\ar[rr,dotted] && X_2 \\ar[ld] \\\\\n& X \\ar[dd] \\\\ \\\\\n& B\n\\end{tikzcd}$$ In this situation, the two-rays game $Y_1/X$ gives a diagram $$\\begin{tikzcd}[link]\nY_1 \\ar[rr, dotted, -] \\ar[dd] && Y_2 \\ar[dd] \\\\ \\\\\nX_1 \\ar[dr] && X' \\ar[dl] \\\\\n& X &\n\\end{tikzcd}$$ where $Y_1 \\ps Y_2$ is a sequence of log-flips and $Y_2 \\to X'$ is a morphism of relative Picard rank $1$, with $X'$ a $\\Q$-factorial variety. ", "If $Y_1/X_1$ is a divisorial contraction, then $Y_2/X'$ must be birational hence also is a divisorial contraction. ", "On the other hand if $Y_1/X_1$ is a Mori fibre space, then $Y_2/X'$ cannot be birational, otherwise $X'/X$ would be a $D$-Mori fibre space for some divisor $D$; impossible since $X'$ is $\\Q$-factorial but not $X$. By uniqueness of the log-flip associated to the small contraction $X_1 \\to X$, we conclude in both cases that $X' = X_2$.\n\nWe now recall the following result of [@Corti1995], which follows from the Negativity Lemma \\[lem:Negativity\\], and then deduce from it Corollaries \\[cor:corti3.5bis\\] and \\[cor:SameDivisors\\], similar to [@Corti1995 Proposition 3.5].", "\n\n\\[lem:corti2.7\\] Let $\\phi\\colon Y \\ps Y'$ be a pseudo-isomorphism between $\\Q$-factorial varieties. ", "If there is an ample divisor $H$ on $Y$ such that $\\phi_*H$ is ample on $Y'$, then $\\phi$ is an isomorphism.", "\n\nIn the following Corollary \\[cor:corti3.5bis\\], the assumption $\\rho(X/B)=1$, which is part of the definition of a Mori fibre space, is crucial. ", "For instance if $\\rho(X/B)=2$, then any Sarkisov diagram whose pseudo-isomorphism on the top row is not an isomorphism provides a counter example (see Definition \\[def:sarkisovLink\\] and Figure \\[fig:SarkisovTypes\\]).", "\n\n\\[cor:corti3.5bis\\] Let $X/B$ and $X'/B$ be Mori fibre spaces over the same base $B$, and $\\phi\\colon X \\ps X'$ a pseudo-isomorphism over $B$, that is, the following diagram commutes: $$\\begin{tikzcd}[link]\nX \\ar[rr,dotted,\"\\phi\" ] \\ar[dr,\"\\eta\",swap] && X' \\ar[dl,\"\\eta'\"] \\\\ \n & B\n\\end{tikzcd}$$ Then $\\phi$ is an isomorphism.", "\n\nLet $A_B, A_X$ be ample divisors respectively on $B$ and $X$, and $0 < \\eps \\ll 1$. If $C$ is a general curve contracted by $X/B$, then $\\phi$ is an isomorphism in a neighborhood of $C$, hence $\\phi_* A_X$ is relatively ample on $X'/B$. Then $\\eta^* A_B + \\eps A_X$ and ${\\eta'}^* A_B + \\phi_*\\eps A_X$ are both ample, and we conclude by Lemma \\[lem:corti2.7\\].", "\n\n\\[cor:SameDivisors\\] Consider a commutative diagram $$\\begin{tikzcd}[link]\nY \\ar[rr,dotted,\"\\phi\" ] \\ar[rd,\"\\pi\",swap] && Y' \\ar[ld,\"\\pi'\"] \\\\\n& X\n\\end{tikzcd}$$ where $X, Y, Y'$ are $\\Q$-factorial varieties, $\\pi$ and $\\pi'$ are divisorial contractions, and $\\phi$ is a pseudo-isomorphism. ", "Then $\\phi$ is an isomorphism.", "\n\nLet $E$ and $E'$ be the exceptional divisors of $\\pi$ and $\\pi'$, respectively. ", "Observe that $\\phi_* E = E'$. Pick $A$ a general ample divisor on $X$ and $0 < \\eps \\ll 1$, and consider $H = \\pi^*A - \\eps E$, $H' = {\\pi'}^*A-\\eps E'$. Both $H$ and $H'$ are ample, and we have $H' = \\phi_* H$, so by Lemma \\[lem:corti2.7\\] we conclude that $Y \\ps Y'$ is an isomorphism.", "\n\n\\[lem:2rays2Divisors\\] Let $T \\to Y$ and $Y\\to X$ be two divisorial contractions between $\\Q$-factorial varieties, with respective exceptional divisors $E$ and $F$. Assume that there exists a morphism $X \\to B$ such that $T/B$ is a Mori dream space. ", "Then there exist two others $\\Q$-factorial varieties $T'$ and $Y'$, with a pseudo-isomorphism $T \\ps T'$ and birational contractions $T' \\to Y' \\to X$, with respective exceptional divisors the strict transforms of $F$ and $E$, such that the following diagram commutes: $$\\begin{tikzcd}[link]\nT \\ar[rr, dotted, -] \\ar[dd,\"E\",swap] && T' \\ar[dd,\"F\"] \\\\ \\\\\nY \\ar[dr,\"F\",swap] && Y' \\ar[dl,\"E\"] \\\\\n& X &\n\\end{tikzcd}$$\n\nThe diagram comes from the two-rays game associated to $T/X$. The only thing to prove is that the divisors are not contracted in the same order on the two sides of the two-rays game. ", "Assume that both $\\pi\\colon Y\\to X$ and $\\pi'\\colon Y' \\to X$ contract the strict transforms of the same divisor $F$. Then $T\\to Y$ and $T'\\to Y'$ both contract a same divisor $E$ and $T\\ps T'$ descends to a pseudo-isomorphism $Y\\ps Y'$. By Corollary \\[cor:SameDivisors\\] the pseudo-isomorphism $Y \\ps Y'$ is an isomorphism. ", "Then applying again Corollary \\[cor:SameDivisors\\] to the two divisorial contractions from $T, T'$ to $Y \\simeq Y'$, with same exceptional divisor $E$, we obtain that $T \\ps T'$ also is an isomorphism. ", "The morphisms $T/Y$ and $T/Y'$ are then divisorial contractions of the same extremal ray, contradicting the assumption that the diagram was produced by a two-rays game.", "\n\nGonality and covering gonality {#sec:gonality}\n------------------------------\n\nRecal that the *gonality* $\\gon(C)$ of a (possibly singular) curve $C$ is defined to be the least degree of the field extension associated to a dominant rational map $C\\rat \\p^1$.\n\nNote that $\\gon(C) = 1$ if and only if $C$ is rational. ", "Moreover, for each smooth curve $C\\subset \\p^2$ of degree $>1$ we have $\\gon(C)=\\deg(C)-1$. Indeed, the inequality $\\gon(C)\\le \\deg(C)-1$ is given by the projection from a general point of $C$ and the other inequality is given by a result of Noether (see for instance [@BDELU]).", "\n\nThe following definitions are taken from [@BDELU] (with a slight change, see Remark \\[rem:CovGonConnGon\\]).", "\n\nFor each variety $X$ we define the *covering gonality* of $X$ to be $$\\covgon(X) = \\min \\left \\{ c > 0 \\ \n\\left | \n\\parbox{2.7in}{\n\\begin{center} \nThere is a dense open subset $U\\subseteq X$ such that each point $x\\in U$ is contained in an irreducible curve $C \\subseteq X$ with $\\gon(C) \\le c$. \n\\end{center}\n}\n\\right \\}\\right..$$ Similarly we define the *connecting gonality* of $X$ to be $$\\conngon(X) = \\min \\left \\{ c > 0 \\ \n\\left | \n\\parbox{2.7in}{\n\\begin{center} \nThere is a dense open subset $U\\subseteq X$ such that any two points $x,y\\in U$ are contained in an irreducible curve $C \\subseteq X$ with $\\gon(C) \\le c$.\n\\end{center}\n}\n\\right \\}\\right..$$\n\n\\[rem:CovGonConnGon\\]\n\n1. ", " Our definitions of the covering and connecting gonality slightly differ from those of [@BDELU], as we ask $\\gon(C)\\le c$ where they ask $\\gon(C)=c$. Lemma \\[lem:Covgonfamily\\] shows that the covering gonality is the same for both definitions. ", "A similar argument should also work for the connecting gonality, but we do not need it here, as we will not use any result of [@BDELU] involving the connecting gonality.", "\n\n2. ", " The covering gonality and connecting gonality are integers which are invariant under birational maps.", "\n\n3. ", " For each variety $X$, we have $$\\covgon(X) \\le \\conngon(X).$$ Moreover, if $\\dim(X)=1$, then $\\covgon(X)=\\conngon(X)=\\gon(X)$.\n\n4. ", " If $\\covgon(X)=1$ one says that $X$ is *uniruled*. ", "This corresponds to asking that the union of all rational curves on $X$ contains an open subset of $X$. Similarly, $X$ is said to be *rationally connected* if $\\conngon(X)=1$. As already mentioned in §\\[sec:moriDream\\], this corresponds to asking that a rational curve passes through two general points.", "\n\n5. ", " \\[gonality:P1xB\\] Each rationally connected variety is uniruled. ", "However, the converse does not hold in general. ", "Indeed, for each variety $B$, we have $\\covgon(B\\times \\p^n)=1$ for each $n\\ge 1$, but $\\conngon(B\\times \\p^n)=\\conngon(B)$ as the following lemma shows: Lemma \\[lem:CovgonRatMorphism\\]\\[covconngonXY\\] applied to $B\\times \\p^n/B$ gives $\\conngon(B\\times \\p^n)\\ge \\conngon(B)$, and the other inequality is given by taking sections in $B\\times \\p^n$ of curves in $B$.\n\nWe recall the following classical facts:\n\n\\[lem:CovgonRatMorphism\\] Let $X,Y$ be varieties and $\\phi\\colon X\\to Y$ a surjective morphism.", "\n\n1. ", " \\[covXY\\] If $X$ and $Y$ have dimension $1$, then $\\gon(X)\\ge \\gon(Y)$.\n\n2. ", " \\[covconngonXY\\] We have $\\conngon(X)\\ge\\conngon(Y)$ $($but not $\\covgon(X)\\ge \\covgon(Y)$ in general, see Remark $\\ref{rem:CovGonConnGon}\\ref{gonality:P1xB})$.\n\n3. ", " \\[covgonfinitemorphism\\] If $\\dim X=\\dim Y$, denote by $\\deg(\\phi)$ the degree of the associated field extension $\\C(Y)\\subseteq \\C(X)$. Then $$\\covgon(X)\\le \\covgon(Y)\\cdot \\deg(\\phi).$$\n\n4. ", " \\[covgondeg\\] If $X\\subseteq\\p^n$ is a closed subvariety, then $\\covgon(X)\\le \\deg(X)$.\n\n\\[covXY\\]. ", "See for instance [@Poonen Proposition A.1(vii)].", "\n\n\\[covconngonXY\\]. ", "We take two general points $y_1,y_2\\in Y$, choose then two general points $x_1,x_2\\in X$ with $\\phi(x_i)=y_i$ for $i=1,2$, and take an irreducible curve $C\\subset X$ of gonality $\\le \\conngon(X)$ which contains $x_1$ and $x_2$. The image is an irreducible curve $\\phi(C)$ of gonality $\\le \\conngon(X)$ (by \\[covXY\\]), containing $y_1$ and $y_2$.\n\n\\[covgonfinitemorphism\\]. ", "By definition of $\\covgon(Y)$, the union of irreducible curves $C$ of $Y$ with $\\gon(C)\\le\\covgon(Y)$ covers a dense open subset of $Y$. Taking the preimages of general such curves, we obtain a covering of a dense open subset of $X$ by irreducible curves $D$ of $X$ with $\\gon(D)\\le \\covgon(Y)\\cdot \\deg(\\phi)$.\n\n\\[covgondeg\\]. ", "If $X\\subseteq\\p^n$ is a closed subvariety, we apply \\[covgonfinitemorphism\\] to the projection onto a general linear subspace $Y\\subseteq\\p^n$ of dimension $\\dim(Y)=\\dim(X)$.\n\n\\[lem:Covgonfamily\\] Let $X$ be a variety with $\\covgon(X)=c$. There is a smooth projective morphism $\\mathcal{C}\\to T$ over a quasi-projective irreducible base variety $T$, with irreducible fibres of dimension one and of gonality $c$, together with a dominant morphism $\\mathcal{C}\\to X$ such that a general fibre of $\\mathcal{C}/T$ is birational to its image in $X$. In particular, there is a dense open subset $U$ of $X$ such that through every point $p\\in U$ there is an irreducible curve $C\\subseteq X$ with $\\gon(C)=c$.\n\nThe proof is analogue to the one of [@GouKou Lemma 2.1]. ", "We consider the Hilbert Scheme $\\Hl$ of all one-dimensional subschemes of $X$, which is not of finite type, but has countably many components. ", "One of the irreducible components contains enough curves of gonality $\\le \\covgon(X)$ to get a dominant map to $X$. We then look at the set of gonality $i$ for each $i$ and obtain algebraic varieties parametrising these, as in [@GouKou Lemma 2.1]. ", "Having finitely many constructible subsets in the image, at least one integer $i\\le \\covgon(X)$ gives a dominant map to $X$ parametrising curves of gonality $i$. By definition of $\\covgon(X)$, this integer $i$ has to be equal to $\\covgon(X)$.\n\nThe following result gives a bound from below that complements the easy bound from above from Lemma \\[lem:CovgonRatMorphism\\].", "\n\n\\[thm:gonality\\] Let $X\\subset \\p^{n+1}$ be an irreducible hypersurface of degree $d\\ge n+2$ with canonical singularities. ", "Then, $ \\covgon(X)\\ge d-n$.\n\nWe now recall the following definition of [@BDELU], which is a birational version of the classical $p$-very ampleness criterion, which asks that every subscheme of length $p+1$ imposes independent conditions on the sections of a line bundle.", "\n\n\\[BVA\\] Let $X$ be variety and let $p\\ge 0$ be an integer.", "\n\nA line bundle $L$ on $X$ *satisfies property $\\BVA_p$* if there exists a proper Zariski-closed subset $Z = Z(L) \\subsetneqq X$ depending on $L$ such that the restriction map $H^{0}(X,L) \\to H^{0}(X,{L \\otimes \\Ol_{\\xi}})$ is surjective for every finite subscheme $\\xi \\subset X$ of length $p+1$ whose support is disjoint from $Z$.\n\nThe line bundle is moreover $p$-*very ample* if one can choose $Z$ to be empty.", "\n\nThis notion is related to the covering gonality via the following result:\n\n\\[thm:BVAK\\] Let $X$ be a variety, and $p\\ge 0$ an integer. ", "If $K_X$ satisfies $\\BVA_p$, then $\\covgon(X)\\ge p+2$.\n\nWe will use the following observations of [@BDELU] to check the hypothesis of Theorem \\[thm:BVAK\\]:\n\n\\[lem:BVApEasy\\] Let $X$ be a variety, $L$ a line bundle on $X$ and $p\\ge 0$ an integer.", "\n\n1. ", " \\[BVAeff\\] If $L$ satisfies $\\BVA_p$ and $E$ is an effective divisor on $X$, then $\\Ol_X(L+E)$ satisfies $\\BVA_p$.\n\n2. ", " \\[BVAmor\\] Suppose that $f\\colon Y\\to X$ is a morphism which is birational onto its image, that $L$ satisfies $\\BVA_p$ and that the closed set $Z\\subseteq X$ from Definition $\\ref{BVA}$ does not contain the image of $f$. Then, $f^*L $ satisfies $\\BVA_p$.\n\n3. ", " \\[BVAPn\\] For each $d\\ge 0$, $\\Ol_{\\p^n}(p)$ is $p$-very ample, i.e. satisfies $\\BVA_p$ with an empty closed set $Z\\subseteq \\p^n$.\n\nThe three assertions follow from the definition of $\\BVA_p$, as mentioned in [@BDELU Example 1.2].", "\n\nRank *r* fibrations and Sarkisov links {#sec:SarkisovLinks}\n======================================\n\nIn this section we introduce the notion of rank $r$ fibration, recovering the notion of Sarkisov link for $r = 2$. Then we focus on rank $r$ fibrations and Sarkisov links with general fibre a curve.", "\n\nRank *r* fibrations\n-------------------\n\nThe notion of rank $r$ fibration is a key concept in this paper. ", "Essentially these are (relative) Mori dream spaces with strong constraints on singularities. ", "The cases of $r = 1,2,3$ will allow us to recover respectively the notion of terminal Mori fibre spaces, of Sarkisov links, and of elementary relations between those. ", "The precise definition is as follows.", "\n\n\\[def:rankFibration\\] Let $r\\ge 1$ be an integer. ", "A morphism $\\eta\\colon X\\to B$ is a *rank $r$ fibration* if the following conditions hold:\n\n1. ", " \\[fib:dream\\] $X/B$ is a Mori dream space (see Definition \\[def:moriDream\\]);\n\n2. ", " \\[fib:eta\\] $\\dim X > \\dim B \\ge 0$ and $\\rho(X/B) = r$;\n\n3. ", " \\[fib:singX\\] $X$ is $\\Q$-factorial and terminal, and for any divisor $D$ on $X$, the output of any $D$-MMP from $X$ over $B$ is still $\\Q$-factorial and terminal (recall that such an output has the same dimension as $X$ by definition, see §\\[sec:MMP\\]);\n\n4. ", " \\[fib:singB\\] $B$ is klt.", "\n\n5. ", " \\[fib:big\\] The anticanonical divisor $-K_X$ is $\\eta$-big (see Lemma \\[lem:piBig\\]).", "\n\nWe say that a rank $r$ fibration $X/B$ *factorises through* a rank $r'$ fibration $X'/B'$, or that *$X'/B'$ is dominated by $X/B$*, if the fibrations $X/B$ and $X'/B'$ fit in a commutative diagram $$\\begin{tikzcd}[link]\nX \\ar[rrr] \\ar[dr,dashed] &&& B \\\\\n& X' \\ar[r] & B' \\ar[ur]\n\\end{tikzcd}$$ where $X \\rat X'$ is a birational contraction, and $B' \\to B$ is a morphism with connected fibres. ", "This implies $r \\ge r'$.\n\n\\[ex:rank\\_r\\]  \n\n1. ", " \\[ex:rank\\_r:2\\] If $X$ is a $\\Q$-factorial terminal Fano variety of rank $r$, then $X/\\pt$ is a rank $r$ fibration. ", "Indeed as already mentioned in Example \\[ex:Mds\\], $X$ is a Mori dream space, and moreover for any divisor $D$ the output of a $D$-MMP is $\\Q$-factorial and terminal. ", "Both assertions follow from the fact that we can pick a small rational number $\\eps >0$ such that $-K_X + \\eps D$ is ample, and then writing $\\eps D = K_X + (-K_X + \\eps D)$ we see that a $D$-MMP is also a ($K_X+$ ample)-MMP.", "\n\n2. ", " \\[ex:rank\\_r:3\\] Let $p_1, p_2$ be two distinct points on a fibre $f$ of $\\p^1 \\times \\p^1/\\p^1$, and consider $S \\to \\p^1 \\times \\p^1$ the blow-up of $p_1$ and $p_2$. Then $S$ is a weak del Pezzo toric surface of Picard rank $4$, hence in particular $S/\\pt$ is a Mori dream space. ", "However $S/\\pt$ is *not* a rank $4$ fibration, because when contracting the strict transform of $f$ one gets a singular point (hence non terminal as we work here with surfaces), which is forbidden by condition \\[fib:singX\\] of Definition \\[def:rankFibration\\].", "\n\nOther basic examples are Mori fibre spaces:\n\n\\[lem:rank1=Mfs\\] Let $\\eta\\colon X \\to B$ a surjective morphism between normal varieties. ", "Then $X/B$ is a rank $1$ fibration if and only if $X/B$ is a terminal Mori fibre space.", "\n\nObserve that if $\\rho(X/B) = 1$, the notions of $\\eta$-ample and $\\eta$-big are equivalent. ", "So the implication $$X/B \\text{ is a rank $1$ fibration} \\implies X/B \\text{ is a Mori fibre space}$$ is immediate from the definitions, and we need to check the converse.", "\n\nAssume that $X/B$ is a Mori fibre space. ", "Then $\\dim X > \\dim B$ and $\\rho(X/B) = 1$, which is \\[fib:eta\\], by Proposition \\[pro:sing\\_of\\_B\\] the base $B$ is klt, which is \\[fib:singB\\], and $-K_X$ is $\\eta$-ample, which gives \\[fib:big\\].", "\n\nWe now prove that $X/B$ is a Mori dream space, which is \\[fib:dream\\]. ", "Condition \\[dream:Q\\_factorial\\] holds by assumption. ", "By Lemma \\[lem:fibresRatConnected\\]\\[connected:-KBigNef\\] the general fibre of $X/B$ is rationally connected with rational singularities, which gives \\[dream:rat\\_connected\\]. ", "Moreover since $\\rho(X/B) = 1$, we have $\\Ample(X/B) = \\Nef(X/B) = \\widebar{\\Mov}(X/B)$ equal to a single ray, and so conditions \\[dream:nef\\] and \\[dream:mov\\] are immediate.", "\n\nFinally we prove \\[fib:singX\\]. ", "By assumption $X$ is terminal and $\\Q$-factorial. ", "For any divisor $D$, either $D$ is $\\eta$-nef and $X/B$ is a $D$-minimal model, or $-D$ is $\\eta$-ample and $X/B$ is a $D$-Mori fibre space. ", "So $X$ is the only possible output for a $D$-MMP, which proves the claim.", "\n\n\\[lem:StableUnderMMP\\] Let $X/B$ be a rank $r$ fibration.", "\n\n1. ", " \\[stab\\_under\\_MMP:1\\] If $X'$ is obtained from $X$ by performing a log-flip $($resp.", " a divisorial contraction$)$ over $B$, then $X'/B$ is a rank $r$ fibration $($resp.", " a rank $(r-1)$-fibration$)$.\n\n2. ", " \\[stab\\_under\\_MMP:2\\] Assume that $X/B$ factorizes through a rank $s$ fibration $X'/B'$ such that the birational map $X \\to X'$ is a morphism. ", "Let $t = \\rho(X/B')$. Then $X/B'$ is a rank $t$ fibration.", "\n\n\\[stab\\_under\\_MMP:1\\]. ", "Let $\\pi\\colon X\\to X'$ be a divisorial contraction over $B$, with exceptional divisor $E$ (the case of a log-flip, which is similar and easier, is left to the reader).", "\n\n\\[fib:dream\\]. ", "The general fibre of $X'/B$ remains rationally connected, and is terminal by Lemma \\[lem:fibresAreTerminal\\], so it remains to show that a Cox sheaf of $X'/B$ is finitely generated (Lemma \\[lem:Mds\\]).", "\n\nLet $L_1, \\dots, L_p \\in \\Pic(X)_\\Q$ and $L_1',\\dots,L_q' \\in \\Pic_\\Q(X')$ such that $\\Eff(X/B) \\subseteq \\sum \\R_+ L_i$ and $\\Eff(X'/B) \\subseteq \\sum \\R_+ L_i'$. For each open set $U \\subseteq B$, by pulling-back we get an injective morphism of algebras $$\\Cox(X'/B;L_1',\\dots,L_q')(U) \\hookto \\Cox(X/B; E,\\pi^*L_1',\\dots,\\pi^*L_q',L_1,\\dots, L_p)(U).$$ Since $X/B$ is a rank $r$ fibration, its Cox sheaf is finitely generated by Lemma \\[lem:Mds\\], and so $\\Cox(X'/B;L_1',\\dots,L_q')$ also is finitely generated by Lemma \\[lem:FiniteGeneration\\].", "\n\n\\[fib:eta\\]. ", "By definition of a divisorial contraction we have $\\dim X' = \\dim X > \\dim B$, and $\\rho(X') = \\rho(X) -1$, so $\\rho(X'/B) = r - 1$.\n\n\\[fib:singX\\]. ", "The output of any MMP from $X'$ also is the output of a MMP from $X$, and so is $\\Q$-factorial and terminal by assumption.", "\n\n\\[fib:singB\\] holds by assumption.", "\n\n\\[fib:big\\]. ", "Follows from the fact that the image of a big divisor by a birational morphism is still big.", "\n\n\\[stab\\_under\\_MMP:2\\]. ", "The conditions of \\[fib:eta\\] and \\[fib:singB\\] hold by assumption. ", "\\[fib:singX\\] follows because any MMP over $B'$ also is a MMP over $B$. For \\[fib:big\\] we observe that a curve contracted by $X/B'$ also is contracted by $X/B$, so a divisor relatively ample for $X/B$ also is relatively ample for $X/B'$. Then we can restrict a decomposition $-K_X = \\eta\\text{-ample} + \\text{effective}$ for $X/B$ to get a similar decomposition for $X/B'$.\n\nFinally we show \\[fib:dream\\]. ", "Let $L_1, \\dots, L_r$ be $\\Q$-divisors on $X$ such that $\\Eff(X/B) \\subseteq \\sum \\R_+ L_i$, which implies $\\Eff(X/B') \\subseteq \\sum \\R_+ L_i$. Let $\\phi\\colon B' \\to B$ the morphism given by assumption. ", "Then for each affine open set $U' \\subset B'$, we have $$\\Cox(X/B'; L_1, \\dots, L_r)(U') = \\Cox(X/B; L_1, \\dots, L_r)(\\phi(U')),$$ and the latter is finitely generated by assumption. ", "A general fibre of $X/B'$ is rationally connected because it is birational to a fibre of $X'/B'$, and it has rational singularities by Lemma \\[lem:fibresAreTerminal\\]. ", "We conclude by Lemma \\[lem:Mds\\].", "\n\n\\[lem:weakFano\\] Any rank $r$ fibration $X/B$ is pseudo-isomorphic, via a sequence of log-flips over $B$, to another rank $r$ fibration $Y/B$ such that $-K_Y$ is relatively nef and big over $B$.\n\nWe run a $(-K)$-MMP from $X$ over $B$ (recall that by Lemma \\[lem:anyMMP\\], one can run a $D$-MMP for an arbitrary divisor $D$). ", "It is not possible to have a divisorial contraction, because by Lemma \\[lem:oppositeSigns\\] the resulting singularity would not be terminal, in contradiction with assumption \\[fib:singX\\] in the definition of rank $r$ fibration. ", "If there exists an extremal class that gives a small contraction, we anti-flip it. ", "After finitely many such steps, either $-K$ is relatively nef, or there exists a fibration such that $K$ is relatively ample. ", "But this last situation contradicts the assumption \\[fib:big\\] that the anti-canonical divisor is big over $B$. So finally $-K$ is also relatively nef over $B$, as expected.", "\n\n\\[cor:weakFanoFiber\\] Let $\\eta\\colon Y \\to B$ be a rank $r$ fibration. ", "Then for a general point $p \\in B$, the fibre $Y_p := \\eta^{-1}(p)$ is pseudo-isomorphic to a weak Fano terminal variety, and the curves in $Y_p$ that are non-positive against the canonical divisor cover a subset of codimension at least $2$ in $Y_p$.\n\nBy Lemma \\[lem:weakFano\\], by performing a sequence of log-flips over $B$, which only affects a general fibre along a codimension 2 subset, we can assume that $-K_Y$ is relatively nef and big over $B$. The fact that $Y_p$ is terminal is Lemma \\[lem:fibresAreTerminal\\]. ", "Let $\\Gamma \\subset Y$ be the subset covered by curves contracted by $Y/B$ that are trivial against the canonical divisor. ", "Then consider the rational map $\\phi := \\lvert -mK_Y \\rvert \\times \\eta \\colon Y \\rat \\p^N \\times B$. By [@Kollar93 Theorem 1.1], $\\phi$ is a morphism, and it is a birational contraction onto its image. ", "If $\\Gamma$ contains a divisor $E$, then $E$ is contracted by $\\phi$, and by Lemma \\[lem:oppositeSigns\\] this would produce a non-terminal singularity, in contradiction with the definition of rank $r$ fibration. ", "So $\\Gamma$ has codimension at least 2 in $Y$, hence $\\Gamma_p = \\Gamma \\cap Y_p$ has codimension at least 2 in $Y_p$ for a general $p$. Since by adjunction $K_{Y_p} = K_Y|_{Y_p}$, $\\Gamma_p$ is exactly the locus of contracted curves in $Y_p$ with trivial intersection against $K_{Y_p}$. The fact that $-K_{Y_p}$ is big over $B$ follows from Lemma \\[lem:piBig\\], by restricting to $Y_p$ a decomposition $-K_Y = \\eta\\text{-ample} + \\text{effective}$.\n\nSarkisov links\n--------------\n\nThe notion of rank $2$ fibration recovers the notion of Sarkisov link:\n\n\\[lem:rank 2 and link\\] Let $Y/B$ be a rank $2$ fibration. ", "Then $Y/B$ factorises through exactly two rank $1$ fibrations $X_1/B_1, X_2/B_2$, which both fit into a diagram $$\\begin{tikzcd}[link]\n\\ar[dd] & Y \\ar[l,dotted] \\ar[r,dotted] & \\ar[dd] \\\\ \\\\\n \\ar[dr] && \\ar[dl] \\\\\n& B &\n\\end{tikzcd}$$ where the top dotted arrows are sequences of log-flips, and the other four arrows are morphisms of relative Picard rank $1$.\n\nThe diagram comes from the two-rays game associated to $Y/B$, as explained in §\\[sec:2rays\\]. ", "Morever, since $\\dim Y > \\dim B$, on each side of the diagram exactly one of the two descending arrows corresponds to a morphisms $X_i \\to B_i$ with $\\dim Y = \\dim X_i > \\dim B_i$. If $B_i = B$ then $X_i/B_i$ is a rank $1$ fibration by Lemma \\[lem:StableUnderMMP\\]\\[stab\\_under\\_MMP:1\\]. ", "If $\\rho(B_i/B) = 1$, we can use Lemma \\[lem:StableUnderMMP\\]\\[stab\\_under\\_MMP:2\\], or alternatively use the following simpler argument. ", "Since $-K_{X_i}$ is relatively big over $B$ we have $-K_{X_i}\\cdot C > 0$ for a general contracted curve of $X_i/B_i$ (write $-K_{X_i} = A + E$ with $A$ relatively ample and $E$ effective, and take $C$ not contained in $E$.) So $-K_{X_i}$ is relatively ample over $B_i$, hence $X_i/B_i$ is a Mori fibre space, or equivalently a rank $1$ fibration (Lemma \\[lem:rank1=Mfs\\]).", "\n\n\\[def:sarkisovLink\\] In the situation of Lemma \\[lem:rank 2 and link\\], we say that the birational map $\\chi\\colon X_1 \\rat X_2$ is a *Sarkisov link*. ", "The diagram is called a *Sarkisov diagram*. ", "Observe that a rank 2 fibration uniquely defines a Sarkisov diagram, but the Sarkisov link is only defined up to taking inverse.", "\n\nIf a rank $r$ fibration factorises through $Y/B$, we equivalently say that it *factorises through* the Sarkisov link associated to $Y/B$.\n\nWe say that the Sarkisov link associated with a rank 2 fibration $Y/B$ is a *Sarkisov link of conic bundles* if $\\dim B = \\dim X - 1$. Observe that in this situation both $X_1/B_1$ and $X_2/B_2$ are indeed conic bundles in the sense of Definition \\[def:conicBundle\\].", "\n\n\\[def:4TypesOfLinks\\] In the diagram of Lemma \\[lem:rank 2 and link\\], there are two possibilities for the sequence of two morphisms on each side of the diagram: either the first arrow is already a Mori fibre space, or it is divisorial and in this case the second arrow is a Mori fibre space. ", "This gives $4$ possibilities, which correspond to the usual definition of *Sarkisov links of type , , and* , as illustrated on Figure \\[fig:SarkisovTypes\\].", "\n\n$${\n\\def\\arraystretch{2.2}\n\\begin{array}{cc}\n\\begin{tikzcd}[ampersand replacement=\\&,column sep=1.3cm,row sep=0.16cm]\n\\ar[dd,\"div\",swap] \\ar[rr,dotted,-] \\&\\& X_2 \\ar[dd,\"fib\"] \\\\ \\\\\nX_1 \\ar[uurr,\"\\chi\",dashed,swap] \\ar[dr,\"fib\",swap] \\& \\& B_2 \\ar[dl] \\\\\n\\& B_1 = B \\&\n\\end{tikzcd}\n&\n\\begin{tikzcd}[ampersand replacement=\\&,column sep=.8cm,row sep=0.16cm]\n\\phantom{X}\\ar[dd,\"div\",swap] \\ar[rr,dotted,-] \\&\\& \\ar[dd,\"div\"] \\\\ \\\\\nX_1 \\ar[rr,\"\\chi\",dashed,swap] \\ar[dr,\"fib\",swap] \\& \\& X_2 \\ar[dl,\"fib\"] \\\\\n\\& B_1 = B = B_2 \\&\n\\end{tikzcd}\n\\\\\n\\I & \\II \n\\\\\n\\begin{tikzcd}[ampersand replacement=\\&,column sep=1.3cm,row sep=0.16cm]\nX_1 \\ar[ddrr,\"\\chi\",dashed,swap] \\ar[dd,\"fib\",swap] \\ar[rr,dotted,-] \\&\\& \\ar[dd,\"div\"] \\\\ \\\\\nB_1 \\ar[dr] \\& \\& X_2 \\ar[dl,\"fib\"] \\\\\n\\& B = B_2 \\&\n\\end{tikzcd}\n&\n\\begin{tikzcd}[ampersand replacement=\\&,column sep=1.7cm,row sep=0.16cm]\nX_1 \\ar[rr,\"\\chi\",dotted,swap] \\ar[dd,\"fib\",swap] \\&\\& X_2 \\ar[dd,\"fib\"] \\\\ \\\\\nB_1 \\ar[dr] \\& \\& B_2 \\ar[dl] \\\\\n\\& B \\&\n\\end{tikzcd}\n\\\\\n\\III & \\IV \n\\end{array}\n}$$\n\n\\[rem:top\\_row\\] The definition of a Sarkisov link in the literature is usually not very precise about the pseudo-isomorphism involved in the top row of the diagram. ", "An exception is [@CPR Definition 3.1.4(b)], but even there they do not make clear that there is at most one flop, and that all varieties admit morphisms to a common $B$.\n\nIn fact, it follows from the definition that there are strong constraints about the sequence of anti-flips, flops and flips (that is, about the sign of the intersection of the exceptional curves against the canonical divisor). ", "Precisely, the top row of a Sarkisov diagram has the following form: $$\\begin{tikzcd}[column sep=1.5cm,row sep=0.2cm]\nY_m \\ar[d] \\ar[r,dotted,<-] & \\dots \\ar[r,dotted,<-] & Y_0 \\ar[r,dotted,-] & Y'_0 \\ar[r,dotted,->] & \\dots \\ar[r,dotted,->] & Y'_n \\ar[d] \\\\\n\\text{} &&&&& \\text{} \\\\\n&& \\ar[r,phantom,\"B\",\"\"{name=B, inner sep=5mm}] &\n\\ar[from=1-1, to=B] \\ar[from=1-2, to=B] \\ar[from=1-3, to=B] \n\\ar[from=1-4, to=B] \\ar[from=1-5, to=B] \\ar[from=1-6, to=B]\n\\ar[from=2-1, to=B] \\ar[from=2-6, to=B]\n\\end{tikzcd}$$ where $Y_0 \\ps Y'_0$ is a flop over $B$ (or an isomorphism), $m,n \\ge 0$, and each $Y_i \\ps Y_{i+1}$, $Y'_i \\ps Y'_{i+1}$ is a flip over $B$. This follows from the fact that for $Y = Y_i$ or $Y_i'$, a general contracted curve $C$ of the fibration $Y/B$ satisfies $K_Y \\cdot C < 0$, hence at least one of the two extremal rays of the cone $\\NE(Y/B)$ is strictly negative against $K_Y$.\n\nObserve also that both $Y_0/B$ and $Y_0'/B$ are relatively weak Fano (or Fano if the flop is an isomorphism) over $B$, as predicted by Lemma \\[lem:weakFano\\]. ", "All other $Y_i/B$ and $Y'_i/B$ are not weak Fano over $B$, but still each is a rank 2 fibration that uniquely defines the Sarkisov diagram.", "\n\n\\[ex:simple\\_links\\] We give some simple examples of Sarkisov links of each type in dimension 3. ", "Here all varieties are smooth, and the pseudo-isomorphisms in the top rows of the Sarkisov diagrams are isomorphisms. ", "For more complicated (and typical) examples, see §\\[sec:examples\\]. ", "Observe that \\[simple\\_link:1\\] and \\[simple\\_link:2\\] are examples of Sarkisov links of conic bundles, while \\[simple\\_link:3\\] and \\[simple\\_link:4\\] are not.", "\n\n1. ", " \\[simple\\_link:1\\] Let $X_1/B_1 = \\p^1 \\times \\p^2 / \\p^2$, and let $X_2 \\to X_1$ be the blow-up of one fibre. ", "Then $X_2 = \\p^1 \\times \\F_1$ is a Mori fibre space over the Hirzebruch surface $B_2 = \\F_1$. The map $\\chi\\colon X_1/B_1 \\rat X_2/B_2$ is a link of type , or equivalently $\\chi^{-1}\\colon X_2/B_2 \\to X_1/B_1$ is a link of type .", "\n\n2. ", " \\[simple\\_link:2\\] Take again $X_1/B_1 = \\p^1 \\times \\p^2 / \\p^2$, let $L \\subset \\p^2$ be a line, and $\\Gamma = \\{0\\} \\times L \\subset X_1$. Let $Y\\to X_1$ be the blow-up of $\\Gamma$, and denote by $D$ the strict transform on $Y$ of $\\p^1 \\times L \\subset X_1$. Then there is a divisorial contraction $Y \\to X_2$ that contracts $D$ to a curve, and $X_2/\\p^2$ is still a $\\p^1$-bundle (but not a trivial product). ", "The map $\\chi\\colon X_1/\\p^2 \\rat X_2/\\p^2$ is a link of type .", "\n\n3. ", " \\[simple\\_link:3\\] A general cubo-cubic map in $\\Bir(\\p^3)$ provides an example of link of type with $X_1, X_2$ equal to $\\p^3$ and $B_1 = B_2 = \\pt$ a point. ", "Indeed the resolution of such a map consists in blowing-up a smooth curve of genus 3 and degree 6 in $X_1$, and then contracting a divisor onto a curve of the same kind in $X_2$. This is the only example of a link of type from $\\p^3$ to $\\p^3$ starting with the blow-up of a smooth curve where the pseudo-isomorphism is in fact an isomorphism: see [@Katz].", "\n\n4. ", " \\[simple\\_link:4\\] Finally, take $X_1 = X_2 = \\p^1 \\times \\p^2$, $B_1 = \\p^1$, $B_2 = \\p^2$, and let $X_1/B_1$ and $X_2/B_2$ be respectively the first and second projection. ", "Then the identity map $\\id\\colon X_1/B_1 \\to X_2/B_2$ is a link of type .", "\n\n\\[lem:distinctDivisors\\] Consider a Sarkisov link of type : $$\\begin{tikzcd}[link]\nY_1 \\ar[rr,dotted,->,\"\\phi\"] \\ar[dd,\"\\pi_1\",swap]&& Y_2 \\ar[dd,\"\\pi_2\"]\\\\ \\\\\nX_1 \\ar[rr,\"\\chi\",dashed] \\ar[rd,swap]&& X_2 \\ar[ld]\\\\\n&B \n\\end{tikzcd}$$ and denote $E_1, E_2$ the respective exceptional divisors of $\\pi_1, \\pi_2$. Then $\\phi_* E_1 \\neq E_2$.\n\nAssume that $\\phi_* E_1 = E_2$. Then $\\chi\\colon X_1 \\ps X_2$ is a pseudo-isomorphism, hence an isomorphism by Corollary \\[cor:corti3.5bis\\]. ", "Then Corollary \\[cor:SameDivisors\\] implies that the pseudo-isomorphism $\\phi\\colon Y_1 \\ps Y_2$ also is an isomorphism. ", "The morphisms $Y_1/X_1$ and $Y_1/X_2$ are then divisorial contractions of the same extremal ray, contradicting the assumption that the diagram was the result of the two-rays game from $Y_1/B$.\n\n\\[lem:B’toB\\] Let $X/B$ be a rank $2$ fibration that factorises through a rank $1$ fibration $\\sigma\\colon X \\to B'$, with $\\dim X - 1 = \\dim B' > \\dim B$. Then $\\eta\\colon B' \\to B$ is a klt Mori fibre space, and in particular for each $p \\in B$, the fibre $\\eta^{-1}(p)$ is covered by rational curves.", "\n\nRecall that $B'$ is klt by \\[fib:singB\\]. ", "We need to show that $-K_{B'}$ is $\\eta$-ample, and then the fibre $\\eta^{-1}(p)$ is covered by rational curves for each $p\\in B$ by Lemma \\[lem:fibresRatConnected\\]\\[connected:-KBigNef\\], applied with $\\Delta = 0$.\n\nBy assumption $\\rho(B'/B) = 1$, so we only need to show that there exists a contracted curve $C \\subseteq B'$ such that $-K_{B'} \\cdot C > 0$. Since $\\dim B' > \\dim B$, the contracted curves cover $B'$, so we can choose $C$ sufficiently general in a fibre $\\eta^{-1}(q)$ of a general point $q\\in B$ such that the following holds:\n\n1. ", " \\[B’toB:1\\] $C$ is not contained in the discriminant locus $\\Delta' \\subset B'$ of the conic bundle $\\sigma\\colon X \\to B'$;\n\n2. ", " \\[B’toB:2\\] The surface $\\sigma^{-1}(C)$ does not contain any of the curves $C' \\subseteq X$ contracted by $\\eta \\circ \\sigma$ with $-K_X \\cdot C' \\le 0$.\n\n3. ", " \\[B’toB:3\\] The fibre $F=(\\eta\\circ \\sigma)^{-1}(q)$ of $\\eta \\circ \\sigma \\colon X \\to B$ containing the surface $\\sigma^{-1}(C)$ is general, so that $(-K_X)|_F$ is big\n\nMore precisely, for \\[B’toB:1\\] is suffices to choose $\\eta^{-1}(q)$ not contained in the hypersurface $\\Delta' \\subset B'$. We can ensure \\[B’toB:2\\] because by Corollary \\[cor:weakFanoFiber\\] such curves cover at most a codimension 2 subset of $F$. Finally for \\[B’toB:3\\] recall first that since $X/B$ is a rank $2$ fibration, $-K_X$ is relatively big by \\[fib:big\\]. ", "Moreover the intersection $(-K_X)|_F \\cdot \\sigma^{-1}(C)$ is a non-trivial effective $1$-cycle. ", "Indeed, since $(-K_X)|_F$ is big, we can take a large integer $m>0$ and find that $(-mK_X)|_F$ induces a rational morphism contracting no curve on the complement of a divisor of $F$. It suffices then to choose $C$ such that $\\sigma^{-1}(C)$ is not contained in this divisor.", "\n\nAs in [@MoriMukai Corollary 4.6], we have $-4K_{B'} \\equiv \\sigma_*(-K_X)^2 + \\Delta$. Intersecting with $C$, we obtain $$\\begin{aligned}\n-4K_{B'}\\cdot C &= \\sigma_*(-K_X)^2 \\cdot C + \\Delta\\cdot C \\\\\n& \\ge (-K_X)|_F \\cdot (-K_X)|_F \\cdot \\sigma^* C \\\\\n& > 0 \\text{ by our choice of $C$}. ", "\\qedhere\\end{aligned}$$\n\nRank *r* fibrations with general fibre a curve {#sec:conicFibrations}\n----------------------------------------------\n\nLet $\\eta\\colon T \\to B$ be a rank $r$ fibration, with $\\dim B = \\dim T -1$. If $\\Gamma \\subset B$ is an irreducible hypersurface, we define $\\eta^\\sharp(\\Gamma) \\subseteq T$ to be the Zariski closure of all fibres of dimension $1$ over $\\Gamma$. The reason for introducing this notion is twofold: first $B$ might not be $\\Q$-factorial, so we cannot consider the pull-back of $\\Gamma$ as a $\\Q$-Cartier divisor, and second the preimage $\\eta^{-1}(\\Gamma)$ might contain superfluous components (see Example \\[ex:type\\_I\\_and\\_II\\_div\\]).", "\n\nNow we distinguish two classes of special divisors in $T$, and we shall show in Proposition \\[pro:rankrFibrations\\] below that they account for the relative rank of $T/B$. Let $D \\subset T$ be a prime divisor. ", "If $\\eta(D)$ has codimension at least 2 in $B$, we say that $D$ is a *divisor of type* . ", "If $\\eta(D)$ is a divisor in $B$, and the inclusion $D \\subsetneq \\eta^{\\sharp} (\\eta (D))$ is strict, we say that $D$ is a *divisor of type* .", "\n\nThe similarity between the terminology for Sarkisov links and for special divisors of type or is intentional. ", "See Lemma \\[lem:LinksCB\\]\\[classification2\\] below.", "\n\n\\[ex:type\\_I\\_and\\_II\\_div\\] We give an example illustrating the definitions above, which also shows that the inclusion $\\eta^\\sharp (\\Gamma) \\subseteq \\eta^{-1}(\\Gamma)$ might be strict. ", "For $B$ an arbitrary smooth variety, consider $Y = \\p^1 \\times B$ with $Y/B$ the second projection. ", "Let $\\Gamma \\subset B$ be any irreducible smooth divisor, $D = \\p^1 \\times \\Gamma$ the pull-back of $\\Gamma$ in $Y$, $\\Gamma' = \\{t\\} \\times \\Gamma \\subset D$ a section and $p \\in D \\setminus \\Gamma'$ a point. ", "Let $T \\to Y$ be the blow-up of $\\Gamma'$ and $p$, with respective exceptional divisors $D'$ and $E$, and denote again $D$ the strict transform of $\\p^1 \\times \\Gamma$ in $T$. Then one can check that the induced morphism $\\eta\\colon T \\to B$ is a rank 3 fibration (see Example \\[ex:with\\_flip\\] for the case $B = \\p^2$), $E$ is a divisor of type I, $D \\cup D'$ is a pair of divisors of type II, and $$\\eta^\\sharp(\\Gamma) = D \\cup D' \\subsetneq D \\cup D' \\cup E = \\eta^{-1}(\\Gamma).$$\n\n\\[pro:rankrFibrations\\] Let $\\eta\\colon T \\to B$ be a rank $r$ fibration, with $\\dim B = \\dim T -1$.\n\n1. ", " \\[rankr:0\\] For any rank $r'$ fibration $T'/B'$ such that $T/B$ factorises through $T'/B'$, any divisor contracted by the birational contraction $T \\rat T'$ is a divisor of type or for $T/B$.\n\n2. ", " \\[rankr:1\\] Divisors of type always come in pairs: for each divisor $D_1$ of type , there exists another divisor $D_2$ of type such that $$D_1 \\cup D_2 = \\eta^{\\sharp} (\\eta (D_1)) = \\eta^{\\sharp} (\\eta (D_2)).$$\n\n3. ", " \\[rankr:2\\] If $D_1 \\cup D_2$ is a pair of divisors of type , and $p$ a general point of $\\eta(D_1) = \\eta(D_2)$, then $\\eta^{-1}(p) = f_1 \\cup f_2$ with $f_i \\subseteq D_i$, $i =1,2$, some smooth rational curves satisfying $$\\begin{aligned}\n K_T \\cdot f_i = -1, && D_i\\cdot f_i = -1, && D_1\\cdot f_2 = D_2\\cdot f_1 = 1.\\end{aligned}$$\n\n4. ", " \\[rankr:3\\] Let $D \\subset T$ be a divisor of type or . ", "Then there exists a birational contraction over $B$ $$T \\rat X \\to B$$ that contracts $D$ and such that $\\rho(X) = \\rho(T) -1$.\n\n5. ", " \\[rankr:4\\] Assume $B$ is $\\Q$-factorial. ", "Let $d_1$ $($resp.", " $d_2)$ be the number of divisors of type $($resp.", " the number of pairs of divisors of type $\\II)$. Then $$r = 1 + d_1 + d_2.$$\n\n\\[rankr:0\\]. ", "Let $D$ be a prime divisor contracted by $T\\rat T'$ and suppose that it is neither of type nor of type for $T/B$. By running a $D$-MMP over $B$ we produce a sequence of log-flips (which do not change the type of special divisors) and then a divisorial contraction. ", "Replacing $T$ by the result of the sequence of log-flips, and $T'$ by the image of the divisorial contraction, we can assume $T \\to T'$ is a divisorial contraction. ", "By Lemma \\[lem:divContToCodim2\\], a general fibre $f$ in the exceptional divisor $D$ is a smooth irreducible rational curve. ", "Now $\\eta(D) \\subset B$ is a divisor because $D$ is not of type , and $D=\\eta^{\\sharp}(\\eta(D))$ because $D$ is a prime divisor and is not of type . ", "But then $f = \\eta^{-1}(p)$ for some $p \\in \\eta(D)$, so $f$ is proportional to a general fibre of $\\eta$, in contradiction with the fact that the extremal contraction of $f$ is divisorial.", "\n\n\\[rankr:1\\] and \\[rankr:2\\]. ", "Let $D_1$ be a divisor of type , and let $D_2, \\dots, D_s$ be the other divisors of type such that $$\\eta^{\\sharp} (\\eta (D_1)) = D_1 \\cup \\dots \\cup D_s.$$ By definition we have $s \\ge 2$, and want to prove $s = 2$. By definition of $\\eta^{\\sharp}$, each $D_i$ is a hypersurface and the general fibres of $D_i\\to \\eta(D_i)$ are curves. ", "Hence, the image of $D_i$ is a hypersurface in $B$. In particular, $\\eta(D_i)=\\eta(D_j)$ for all $i,j\\in \\{1,\\ldots,s\\}$. Let $p \\in \\eta (D_{1})$ be a general point, and write $\\eta^{-1} (p) = f_1 \\cup \\dots \\cup f_s$ with $f_i$ a curve in $D_i$. Let $f$ be a general fibre of $\\eta$. We have $D_i \\cdot f = 0$ for each $i$, $D_i \\cdot f_j > 0$ for at least one $j$ (because $\\eta^{-1} (p)$ is connected) and $f \\equiv f_1 + \\dots + f_s$, which gives $$D_i \\cdot f_i < 0.$$ Then by running a $D_i$-MMP from $T$ over $B$, one obtains a sequence of log-flips that does not affect the general fibre $\\eta^{-1} (p)$, and then a divisorial contraction between $\\Q$-factorial and terminal varieties, with exceptional divisor $D_i$ and center of codimension $2$. By Lemma \\[lem:divContToCodim2\\], this implies that $f_i$ is smooth with $K_T \\cdot f_i = D_i \\cdot f_i = -1$. But $K_T \\cdot f = -2$, so we conclude that $s = 2$ as expected. ", "The equality $D_1 \\cdot f_2 = 1$ (and similarly $D_2 \\cdot f_1 = 1$) follows from $D_1 \\cdot f = 0$, $f \\equiv f_1 + f_2$ and $D_1 \\cdot f_1 = -1$.\n\nTo prove \\[rankr:3\\], it suffices to show that the divisor $D$ is covered by curves $\\ell$ such that $D \\cdot \\ell <0$, since then we can get the expected birational contraction by running a $D$-MMP. ", "When $D$ has type we already showed in \\[rankr:2\\] that $D$ is covered by such curves. ", "Now let $D$ be a divisor of type , $p$ a general point in $\\eta(D)$, and let $d \\ge 0$ be the dimension of $\\eta(D)$. By definition of a divisor of type we have $n -3 \\ge d$, where $n = \\dim T$ (In particular divisors of type do not exist on surfaces). ", "Now consider the surface $S \\subset T$ obtained by taking the following intersection of $n-2$ divisors: $$S = \\bigcap_{i = 1}^{n-2-d} H_i \\cap \\bigcap_{j = 1}^{d} \\eta^* H'_j,$$ where the $H_i$ are general hyperplane sections of $T$, and the $H'_j$ general hyperplane sections of $B$ through $p$. By construction, one of the irreducible components of $S \\cap D$ is a curve $\\ell$ contracted to $p$ by $\\eta$. Moreover $\\eta(S)$ is a surface; indeed the general fibres of $\\eta$ are curves and $H_1$ is transverse to the fibres so the morphism from $H_1$ to $\\eta(H_1)$ has finite fibres. ", "The same then holds for $S$, as $n-2-d\\ge 1$. We obtain $D \\cdot \\ell = (\\ell \\cdot \\ell)_S < 0$ as expected (a curve contracted by a morphism between two surfaces has negative self-intersection).", "\n\nTo prove \\[rankr:4\\], observe first that the contraction of a divisor of type does not affect the other divisors of type or , and the contraction of a divisor of type only affects the other divisor in the pair, which is not a divisor of type anymore. ", "So after applying several times \\[rankr:3\\], we may assume $d_1 = d_2 = 0$, and we want to show $r = 1$, or equivalently, that $T/B$ is a Mori fibre space. ", "Then we run a MMP from $T$ over $B$. A flip does not change $d_1$ nor $d_2$, so we can assume that we have a divisorial contraction or a Mori fibre space. ", "A divisorial contraction would produce a divisor of type or by \\[rankr:0\\] (depending on the codimension of the centre), in contradiction with our assumption $d_1 = d_2 = 0$. On the other hand, if $T \\to B'$ is a Mori fibre space, then both $B'$ and $B$ are $(n-1)$ dimensional varieties, and $B'$ is $\\Q$-factorial klt by Proposition \\[pro:sing\\_of\\_B\\]. ", "If the birational morphism $B' \\to B$ is not an isomorphism, it must contracts at least one divisor $D$ because $B$ is $\\Q$-factorial by assumption. ", "By Lemma \\[lem:B’\\_over\\_B\\_is\\_Mds\\] $B'/B$ is a Mori dream space, so we can run a $D$-MMP from $B'$ over $B$. After a sequence of $D$-flips this has to produce a divisorial contraction, hence a divisor of type in $T$ by pulling-back, and again a contradiction. ", "In conclusion, $B' \\iso B$ is an isomorphism and we have a Mori fibre space $T/B$, as expected.", "\n\n\\[lem:gonalityOfTypeII\\] Let $\\eta\\colon T \\to B$ be a rank $r$ fibration with $\\dim B = \\dim T -1$. Assume that $D$ is a divisor of type for $T/B$, with $\\covgon(\\eta(D))> 1$. Then for any rank $r'$ fibration $T'/B'$ that factorises through $T/B$, with $\\dim B'=\\dim T'-1=\\dim B$, the strict transform of $D$ is a divisor of type for $T'/B'$.\n\nRecall that $T'\\rat T$ is a birational contraction and $\\pi\\colon B\\to B'$ is a morphism with connected fibres between klt varieties (Definition \\[def:rankFibration\\]), which in our situation is birational as $\\dim(B)=\\dim(B')$. We write $D=D_1$ and by Proposition \\[pro:rankrFibrations\\]\\[rankr:1\\] we have a pair $D_1\\cup D_2$ of divisors of type for $T/B$, where $\\Gamma=\\eta(D_1)=\\eta(D_2)$ is a divisor of $B$ and $D_1 \\cup D_2 = \\eta^{\\sharp} (\\Gamma)$.\n\nWe first observe that the image of $\\Gamma$ in $B'$ is again a divisor $\\Gamma'\\subset B'$. Indeed otherwise, the divisor $\\Gamma \\subset B$ is one of the divisors contracted by the birational morphism $\\pi\\colon B \\to B'$. By Lemma \\[lem:fibresRatConnected\\]\\[connected:birational\\], this implies that $\\Gamma$ is covered by rational curves, in contradiction with our assumption $\\covgon(\\Gamma) > 1$.\n\nWriting $\\eta'\\colon T'\\to B'$ the rank $r'$ fibration, one observe that the strict transforms $\\tilde{D}_1$ and $\\tilde{D}_2$ of $D_1$ and $D_2$ are such that $\\tilde{D}_1\\cup \\tilde{D}_2\\subseteq \\eta^{\\sharp}(\\Gamma')$. Hence, $\\tilde{D}_1$ and $\\tilde{D}_2$ are divisors of type for $T'/B'$.\n\n\\[lem:factorThroughRank1\\] Let $T/B$ be a rank $r$ fibration with $\\dim B = \\dim T -1$ and $B$ $\\Q$-factorial. ", "Assume that for each divisor $D$ of type for $T/B$, we have $$\\covgon(\\eta(D)) > 1.$$ Then $T/B$ factorises through a rank $1$ fibration $T'/B'$ such that $T \\ps T'$ is a pseudo-isomorphism if and only if $T/B$ does not admit any divisor of type .", "\n\nIf this holds, then $\\dim B' = \\dim T - 1$, $B' \\to B$ is a birational morphism and $\\rho(B'/B) = r-1$.\n\nIf $T/B$ factorises through a rank $r'$ fibration $T'/B'$ such that $T \\ps T'$ is a pseudo-isomorphism, first observe that $\\rho(B'/B) = r - r'$, and $B'\\to B$ is birational, since $\\dim(B)=\\dim(B')$, which follows from $$\\dim(T)=\\dim(T')>\\dim(B')\\ge\\dim(B)=\\dim(T)-1.$$\n\nIf $D_1\\cup D_2$ is a pair of divisors of type for $T/B$, then their strict transforms $\\tilde{D}_1,\\tilde{D}_2$ have the same image in $B'$, which is a divisor because $B'\\to B$ is birational. ", "So if $T/B$ admits at least one divisor of type , then by Proposition \\[pro:rankrFibrations\\]\\[rankr:2\\] some fibres of $T'/B'$ have the form $f_1 + f_2$ with $f_1, f_2$ non proportional. ", "In particular $r' = \\rho(T'/B') \\ge 2$ and so $T'/B'$ is not a Mori fibre space.", "\n\nTo prove the converse, we assume that $T/B$ does not admit any divisor of type , and we proceed by induction on the number $d_1$ of divisors of type . ", "If $d_1 = 0$ then by Proposition \\[pro:rankrFibrations\\]\\[rankr:4\\], $T/B$ is already a rank $1$ fibration, so we just take $T'/B' = T/B$. Now if $d_1 > 0$, by Proposition \\[pro:rankrFibrations\\]\\[rankr:3\\] there exists a birational contraction over $B$, $T \\rat X_1 \\to B$, which contracts one divisor $D$ of type . ", "Since the contraction is obtained by running a $D$-MMP, in fact it factorises as $T \\ps T_1 \\to X_1$, where $T \\ps T_1$ is a sequence of $D$-flips and $T_1 \\to X_1$ is a divisorial contraction. ", "Then by induction hypothesis $X_1/B$ factorises through a rank $1$ fibration $X_2/B_2$ with $X_1 \\ps X_2$ a pseudo-isomorphism (here we use Lemma \\[lem:gonalityOfTypeII\\], which shows that $X_1/B$ does not admit any divisor of type ). ", "By Lemma \\[lem:2raysOverFlip\\], there exist a pseudo-isomorphism $T_1 \\ps T_2$ and a divisorial contraction $T_2 \\to X_2$ that makes the diagram on Figure \\[fig:contractsTypeI\\] commute. ", "Finally we play the two-rays game $T_2/B_2$. Since $T_2/B_2$ admits one divisor of type and no divisor of type (by our assumption on the covering gonality and by Lemma \\[lem:gonalityOfTypeII\\]), the other side of the two-rays game must be a Mori fibre space, which gives the expected rank $1$ fibration $T'/B'$.\n\n$$\\begin{tikzcd}[link]\nT \\ar[ddddddr] \\ar[r,dotted] & T_1 \\ar[r,dotted] \\ar[dd] & T_2 \\ar[r,dotted] \\ar[dd] & T ' \\ar[dd] \\\\ \\\\\n & X_1 \\ar[dddd] \\ar[r,dotted] & X_2 \\ar[dd] & B' \\ar[ldd] \\\\ \\\\\n & & B_2 \\ar[ddl] \\\\ \\\\\n & B\n\\end{tikzcd}$$\n\nSarkisov links of conic bundles {#sec:SarkisovLinksCB}\n-------------------------------\n\nIn this subsection, by applying Proposition \\[pro:rankrFibrations\\] to the case $r=2$, we classify Sarkisov links of conic bundles.", "\n\n\\[lem:LinksCB\\] Let $Y/B$ be a rank $2$ fibration with $\\dim B = \\dim Y -1$, and $\\chi$ the associated Sarkisov link, well-defined up to taking inverse.", "\n\n1. ", " \\[classification1\\] $\\chi$ has type if and only if $B$ is not $\\Q$-factorial.", "\n\n2. ", " \\[classification2\\] If $B$ is $\\Q$-factorial, let $d_1$ $($resp.", " $d_2)$ be the number of special divisors of type $($resp.", " of type $)$ for $Y/B$. Then\n\n - $\\chi$ has type or if and only if $(d_1,d_2)=(1,0)$.\n\n - $\\chi$ has type if and only if $(d_1,d_2)=(0,1)$.\n\n\\[classification1\\]. ", "If $B$ is not $\\Q$-factorial, then it follows directly that $\\chi$ has type , from the fact that the base of a terminal Mori fibre space if always $\\Q$-factorial (Proposition \\[pro:sing\\_of\\_B\\]), and by inspection of the diagrams in Figure \\[fig:SarkisovTypes\\]. ", "Conversely, assuming that $\\chi\\colon X_1/B_1 \\rat X_2/B_2$ is a link of type , we show that $B$ is not $\\Q$-factorial. ", "As $\\dim B=\\dim Y-1$, the morphisms $B_1/B$, $B_2/B$ are birational. ", "If $B$ is $\\Q$-factorial, then $B_1/B$ and $B_2/B$ are birational contractions with respective exceptional divisors $E_1$ and $E_2$. If the birational map $B_1 \\rat B_2$ sends $E_1$ onto $E_2$, then the map is a pseudo-isomorphism, hence an isomorphism by Corollary \\[cor:SameDivisors\\], and then $X_1 \\ps X_2$ also is an isomorphism by Corollary \\[cor:corti3.5bis\\], a contradiction. ", "Otherwise, the pull-backs of $E_1$, $E_2$ together with the choice of any ample divisor give three independent classes in $N^1(Y/B)$, in contradiction with $\\rho(Y/B) = 2$.\n\nTo prove \\[classification2\\], first we observe that Proposition \\[pro:rankrFibrations\\]\\[rankr:4\\] gives $d_1 + d_2 = 1$, hence the two possibilities $(d_1,d_2) = (1,0)$ or $(0,1)$. Recall also from Proposition \\[pro:rankrFibrations\\]\\[rankr:0\\] that any divisor contracted by a birational contraction from $Y$ over $B$ must be of type or . ", "If the link $\\chi$ is of type , then Lemma \\[lem:distinctDivisors\\] gives two birational contractions from $Y$ contracting distinct prime divisors, and this is possible only in the case $(d_1,d_2) = (0,1)$ where there is a pair of divisors of type available. ", "Conversely, if $(d_1,d_2)=(0,1)$, we have two distinct prime divisors, that we can contract via two distinct birational contractions (Proposition \\[pro:rankrFibrations\\]\\[rankr:3\\]). ", "These are the two starting moves of a $2$-ray game which provides a link of type .", "\n\n\\[cor:SarkiIConic\\] Let $\\chi$ be a Sarkisov link of conic bundles of type : $$\\begin{tikzcd}[link]\nY_1 \\ar[dd,\"\\pi_1\",swap] \\ar[rr,dotted,-] && X_2 \\ar[dd,\"\\eta_2\"] \\\\ \\\\\nX_1 \\ar[uurr,\"\\chi\",dashed,swap] \\ar[dr,\"\\eta_1\",swap] & & B_2 \\ar[dl] \\\\\n& B_1 &\n\\end{tikzcd}$$ Let $E_1$ be the exceptional divisor of the divisorial contraction $\\pi_1$. Then $\\eta_1 \\circ \\pi_1(E_1)$ has codimension at least $2$ in $B_1$.\n\nFollows from the fact that $E_1$ is a divisor of type for $Y_1/B_1$.\n\n\\[rem:TypeIVhigher\\] There are examples of link of type as in Lemma \\[lem:LinksCB\\]\\[classification1\\] only when $\\dim B \\ge 3$, hence $\\dim Y \\ge 4$. See the discussion on the two subtypes of type IV links in [@HMcK p. 391 after Theorem 1.5]. ", "For instance, take $B_1$ and $B_2$ that differ by a log-flip, and $B$ the non $\\Q$-factorial target of the associated small contractions. ", "Then the birational map from $(\\p^1\\times B_1)/B_1$ to $(\\p^1\\times B_2)/B_2$ induced by the log-flip is a link of type .", "\n\nNow we focus on the case of Sarkisov links of conic bundles of type . ", "First we introduce the following definition.", "\n\n\\[def:markedConic\\] A *marked conic bundle* is a triple $(X/B,\\Gamma)$, where $X/B$ is a conic bundle in the sense of Definition \\[def:conicBundle\\], and $\\Gamma \\subset B$ is an irreducible hypersurface, not contained in the discriminant locus of $X/B$ $($i.e. the fibre of a general point of $\\Gamma$ is isomorphic to $\\p^1)$. The *marking* of the marked conic bundle is defined to be $\\Gamma$.\n\nWe say that two marked conic bundles $(X/B,\\Gamma)$, and $(X'/B',\\Gamma')$ are *equivalent* if there exists a commutative diagram $$\\begin{tikzcd}[link]\nX \\ar[dd,swap]\\ar[r,\"\\psi\",dashed]& X'\\ar[dd] \\\\ \\\\\nB \\ar[r,\"\\theta\",dashed]& B' \n\\end{tikzcd}$$ where $\\psi, \\theta$ are birational and such that the restriction of $\\theta$ induces a birational map $\\Gamma\\rat \\Gamma'$ between the markings. ", "In particular, if $(X/B,\\Gamma)$, and $(X'/B',\\Gamma')$ are equivalent, then the conic bundles $X/B$ and $X'/B'$ are equivalent in the sense of Definition \\[def:conicBundle\\].", "\n\nFor each variety $Z$, we denote by $\\CB(Z)$ the set of equivalence classes of conic bundles $X/B$ with $X$ birational to $Z$ and denote, for each class of conic bundles $C\\in \\CB(Z)$ by $\\MC(C)$ the set of equivalence classes of marked conic bundles $(X/B,\\Gamma)$ where $C$ is the class of $X/B$.\n\nThe next lemma explains how a Sarkisov link of conic bundles of type gives rise to an equivalence class of marked conic bundles.", "\n\n\\[lem:SarkiIIConic\\] Let $\\chi$ be a Sarkisov link of conic bundles of type between varieties of dimension $n \\ge 2$. Recall that $\\chi$ fits in a commutative diagram of the form $$\\begin{tikzcd}[link]\nY_1 \\ar[rr,dotted,\"\\phi\"] \\ar[dd,\"\\pi_1\",swap]&& Y_2 \\ar[dd,\"\\pi_2\"]\\\\ \\\\\nX_1 \\ar[rr,\"\\chi\",dashed] \\ar[rd,\"\\eta_1\",swap]&& X_2 \\ar[ld,\"\\eta_2\"]\\\\\n&B \n\\end{tikzcd}$$ where $X_1, X_2, Y_1, Y_2$ are $\\Q$-factorial terminal varieties of dimension $n$, $B$ is a $\\Q$-factorial klt variety of dimension $n-1$, $\\phi$ is a sequence of log-flips over $B$, and each $\\pi_i$ is a divisorial contraction with exceptional divisor $E_i \\subset Y_i$ and centre $\\Gamma_i = \\pi_i(E_i) \\subset X_i$.\n\nThen there exists an irreducible hypersurface $\\Gamma \\subset B$ $($of dimension $n-2)$ such that\n\n1. ", " \\[SarkiII:1\\] for $i=1,2$, the centre $\\Gamma_i = \\pi_i(E_i)$ has codimension $2$ in $X_i$, and the restriction $\\eta_i|_{\\Gamma_i}\\colon \\Gamma_i \\to \\Gamma$ is birational. ", "In particular, for each $i$ we have $\\eta_i \\circ \\pi_i (E_i) = \\Gamma$, and the marked conic bundles $(X_1/B,\\Gamma)$ and $(X_2/B,\\Gamma)$ are equivalent.", "\n\n2. ", " \\[SarkiII:4\\] Let $Y$ be equal to $Y_1$, $Y_2$, or any one of the intermediate varieties in the sequence of log-flips $\\phi$. Then $E_1\\cup E_2$ is a pair of divisors of type for $Y/B$.\n\n3. ", " \\[SarkiII:2\\] $\\Gamma$ is not contained in the discriminant locus of $\\eta_1$, or equivalently of $\\eta_2$, which means that a general fibre of $\\eta_i\\colon \\eta_i^{-1}(\\Gamma)\\to \\Gamma$ is isomorphic to $\\p^1$.\n\n4. ", " \\[SarkiII:3\\] At a general point $x \\in \\Gamma_i$, the fibre of $X_i/B$ through $x$ is transverse to $\\Gamma_i$.\n\n\\[SarkiII:1\\] and \\[SarkiII:4\\]. ", "By Lemma \\[lem:LinksCB\\], $Y_1/B$ admits no divisor of type , and exactly one pair of divisors of type . ", "By Lemma \\[lem:distinctDivisors\\] we have $\\phi_* E_1 \\neq E_2$, so the birational contractions $Y_1 \\rat X_1$ and $Y_1 \\rat X_2$ contract distinct divisors. ", "It follows from Proposition \\[pro:rankrFibrations\\] that the pair of divisors of type is $E_1 \\cup E_2$. So by definition $E_1$ and $E_2$ projects to the same hypersurface $\\Gamma \\subset B$. By Proposition \\[pro:rankrFibrations\\]\\[rankr:2\\] both finite maps $\\Gamma_i\\to \\Gamma$ are birational, otherwise the fibre in $Y_i$ over a general point of $\\Gamma$ would have more than two components.", "\n\n\\[SarkiII:2\\] and \\[SarkiII:3\\] follow from Proposition \\[pro:rankrFibrations\\]\\[rankr:2\\]. ", "Indeed if $\\Gamma$ was in the discriminant locus of $\\eta_1$ then the preimage in $Y_1$ of a general point $p \\in B$ would have 3 irreducible components, instead of 2. ", "Moreover writing $f_1 \\cup f_2$ the fibre through $x$, with $f_i \\subseteq E_i$, the fact that the fibre is transverse to $\\Gamma_i$ is equivalent to $f_1 \\cdot E_2 = f_2 \\cdot E_1 = 1$.\n\nBy Lemma \\[lem:SarkiIIConic\\]\\[SarkiII:1\\], to each Sarkisov link of conic bundles of type $\\chi\\colon X_1\\rat X_2$, we can associate the equivalence class of the marked conic bundle $(X_1/B,\\Gamma)$ given in this lemma. ", "We define the *marking* of $\\chi$ to be $\\Gamma\\subset B$. We say that two Sarkisov links of conic bundles of type are *equivalent* if their corresponding marked conic bundles are equivalent.", "\n\nWe also extend the notion of covering gonality (see §\\[sec:gonality\\]) to Sarkisov links of conic bundles of type .", "\n\n\\[def:Covgonchi\\] Let $\\chi$ be a Sarkisov of conic bundles of type between varieties of dimension $n\\ge 3$. We define $\\covgon(\\chi)$ to be $\\covgon(\\Gamma)$, where $\\Gamma$ is the marking of $\\chi$.\n\nIf two Sarkisov links of conic bundles of type are equivalent, then their markings are birational to each other. ", "In particular the number $\\covgon(\\chi)$ only depends on the equivalence class of $\\chi$.\n\nThe above definition makes sense if the varieties $X_i$ have dimension $\\ge 2$, but it is not a very good invariant if the dimension is $2$, as the centre is always a point, and there is only one class of marked conic bundles, given by a point in the base of a Hirzebruch surface. ", "However, the analogue definition over $\\Q$ or over a finite field, instead of over $\\C$, is interesting even for surfaces.", "\n\nRelations between Sarkisov links {#sec:RelSarkisov}\n================================\n\nThe fact that one can give a definition of Sarkisov links in terms of relative Mori dream spaces of Picard rank 2 as in the previous section was independently observed in [@AZ §2] and [@LZ §2.3]. ", "Our next aim is to extend this observation to associate some relations between Sarkisov links to each rank 3 fibration. ", "First we define elementary relations, and then we relate this notion to the work of A.-S. Kaloghiros about relations in the Sarkisov programme.", "\n\nElementary relations\n--------------------\n\n\\[def:Tequivalent\\] Let $X/B$ and $X'/B'$ be two rank $r$ fibrations, and $T \\rat X$, $T \\rat X'$ two birational maps from the same variety $T$. We say that $X/B$ and $X'/B'$ are *$T$-equivalent* (the birational maps being implicit) if there exist a pseudo-isomorphism $X \\ps X'$ and an isomorphism $B \\iso B'$ such that all these maps fit in a commutative diagram: $$\\begin{tikzcd}[link]\n & T \\ar[dl, dashed] \\ar[dr, dashed] \\\\\nX \\ar[dd] \\ar[rr,dotted] & & X' \\ar[dd] \\\\ \\\\\nB \\ar[rr,\"\\sim\"] & & B' \\\\ \n\\end{tikzcd}$$\n\n\\[lem:2 rank 2\\] Let $X_3/B_3$ be a rank $3$ fibration that factorises through a rank $1$ fibration $X_1/B_1$. Then up to $X_3$-equivalence there exist exactly two rank $2$ fibrations that factorise through $X_1/B_1$, and that are dominated by $X_3/B_3$.\n\nWe distinguish three cases according to $\\rho(B_1/B_3)$.\n\nIf $\\rho(B_1/B_3) = 2$, then $B_1$ – being the base of a Mori fibre space – is $\\Q$-factorial klt (Proposition \\[pro:sing\\_of\\_B\\]), and $B_1/B_3$ is a Mori dream space by Lemma \\[lem:B’\\_over\\_B\\_is\\_Mds\\]. ", "The associated two-rays game yields exactly two non-isomorphic $B_2, B_2'$ with $\\rho(B_2/B_3) = \\rho(B_2'/B_3) = 1$. Then Lemma \\[lem:2raysOverFlip\\] provides sequences of log-flips over $B_3$, $X_1 \\ps X_2$ and $X_1 \\ps X_2'$, such that $X_2/B_2$, $X_2'/B_2'$ are the expected rank 2 fibrations.", "\n\nIf $\\rho(B_1/B_3) = 1$, then the base $B_2$ of any of the expected rank 2 fibrations must be equal to $B_1$ or $B_3$, because by assumption we have morphisms $B_1 \\to B_2 \\to B_3$. By Lemma \\[lem:StableUnderMMP\\]\\[stab\\_under\\_MMP:1\\] $X_1/B_3$ is the first expected rank 2 fibration, and up to equivalence it is the only one with base $B_3$, because any rank 2 fibration $X_2/B_3$ satisfies $\\rho(X_2) = \\rho(X_1)$, so the birational contraction $X_2 \\rat X_1$ is a pseudo-isomorphism. ", "Let $D$ be the pull-back on $X_3$ of an ample divisor on $X_1$. The birational contraction $X_3 \\rat X_1$ is a $D$-MMP over $B_3$, and as $\\rho(X_3)-\\rho(X_1)=1$, it decomposes as a sequence of $D$-flips $X_3 \\ps X_3'$, a divisorial contraction $X_3' \\to X_1'$, and a sequence of $D$-flips $X_1' \\ps X_1$. Then Lemma \\[lem:2raysOverFlip\\] provides a sequence of log-flips over $B_3$, $X_3' \\ps X_2$, such that $X_2\\to X_1$ is a divisorial contraction, and by Lemma \\[lem:StableUnderMMP\\] $X_2/B_1$ is the second expected rank 2 fibrations. ", "Any other rank 2 fibration $X_2'/B_1$ satisfying the lemma is equivalent to $X_2/B_1$, because as before the condition on Picard numbers forces $X_2 \\rat X_2'$ to be a pseudo-isomorphism.", "\n\nIf $\\rho(B_1/B_3) = 0$, then $\\rho(X_3) - \\rho(X_1) = 2$, and $B_1 = B_3$ must be the base of any of the expected rank 2 fibrations. ", "By applying several times Lemma \\[lem:2raysOverFlip\\] we construct a sequence of log-flips over $B_3$, $X_3 \\ps X_3'$, such that $X_3' \\to X_1$ is a morphism. ", "The associated two-rays game yields exactly two divisorial contractions $X_2 \\to X_1$ and $X_2' \\to X_1$. Moreover $X_2$ and $X_2'$ are not pseudo-isomorphic by Lemma \\[lem:2rays2Divisors\\], and are uniquely determined up to equivalence by Corollary \\[cor:SameDivisors\\]. ", "Then $X_2/B_1$ and $X_2'/B_1$ are the expected rank 2 fibrations.", "\n\n\\[pro:from T3\\] Let $T/B$ be a rank $3$ fibration. ", "Then there are only finitely many Sarkisov links $\\chi_i$ dominated by $T/B$, and they fit in a relation $$\\chi_t \\circ \\dots \\circ \\chi_1 = \\id.$$\n\nSince $T/B$ is a Mori dream space, by Lemma \\[lem:anyMMP\\] there are only finitely many rank $1$ or 2 fibrations dominated by $T/B$. We construct a bicolored graph $\\Gamma$ as follows. ", "Vertices are rank $1$ or 2 fibrations dominated by $T/B$, up to $T$-equivalence, and we put an edge between $X_2/B_2$ and $X_1/B_1$ if $X_2/B_2$ is a rank 2 fibration that factorises through the rank $1$ fibration $X_1/B_1$. By construction, two vertices of rank $1$ of $\\Gamma$ are at distance 2 if and only if there is a Sarkisov link between them. ", "Then by Lemmas \\[lem:rank 2 and link\\] and \\[lem:2 rank 2\\] we obtain that $\\Gamma$ is a circular graph, giving the expected relation.", "\n\n\\[def:elementary relation\\] In the situation of Proposition \\[pro:from T3\\], we say that $$\\chi_t \\circ \\dots \\circ \\chi_1 = \\id$$ is an *elementary relation* between Sarkisov links, coming from the rank 3 fibration $T/B$. Observe that the elementary relation is uniquely defined by $T/B$, up to taking the inverse, cyclic permutations and insertion of isomorphisms.", "\n\nGeography of ample models {#sec:geography}\n-------------------------\n\nIn this section, we recall some preliminary material from [@BCHM; @HMcK]. ", "The aim is to explain the construction of a polyhedral complex attached with the choice of some ample divisors on a smooth variety, and to state some properties (Proposition \\[pro:coneC\\] and Lemma \\[lem:faces\\_factory\\]) that we will use in the next section to understand relations between Sarkisov links.", "\n\n\\[def:amplesemiample\\] Let $Z$ be a terminal $\\Q$-factorial variety, $D$ be a $\\R$-divisor on $Z$ and $\\phi\\colon Z\\dasharrow Y$ a dominant rational map to a normal variety $Y$. We take a resolution $$\\begin{tikzcd}[link]\n& W \\ar[dl,\"p\",swap] \\ar[dr,\"q\"]\\\\\nZ \\ar[rr, dashed,\"\\phi\"]& &Y\n\\end{tikzcd}$$ where $W$ is smooth, $p$ is a birational morphism and $q$ a morphism with connected fibres. ", "We say that *$\\phi$ is an ample model of $D$* if there exists an ample divisor $H$ on $Y$ such that $p^*D$ is linearly equivalent to $q^*H+E$ where $E\\ge 0$, and if for each effective $\\R$-divisor $R$ linearly equivalent to $p^*D$ we have $R\\ge E$.\n\nIf $\\phi$ is a birational contraction, we say that $\\phi$ is *a semiample model of $D$* if $H=\\phi_*D$ is semiample (hence in particular $\\R$-Cartier) and if $p^*D=q^*H+E$ where $E\\ge 0$ is $q$-exceptional.", "\n\n\\[lem:semiampleample\\] Let $Z$ be a terminal $\\Q$-factorial variety and $D$ a $\\R$-divisor on $Z$.\n\n1. ", " \\[uniqueAmple\\] If $\\phi_i\\colon Z\\dasharrow Y_i$, $i=1,2$, are two ample models of $D$, there exists an isomorphism $\\theta\\colon Y_1\\iso Y_2$ such that $\\phi_2=\\theta\\circ \\phi_1$.\n\n2. ", " \\[SemiampleFactorisation\\] If a birational map $\\psi\\colon Z\\dasharrow X$ is a semiample model of $D$, the ample model $\\phi\\colon Z\\dasharrow Y$ exists and $\\phi=\\theta\\circ \\psi$ for some morphism $\\theta\\colon X\\to Y$. Moreover, $\\psi_* D =\\theta^* H$, where $H$ is the ample divisor $H=\\phi_* D$.\n\n3. ", " \\[birsemiampleample\\] A birational map $\\phi\\colon Z\\dasharrow Y$ is the ample model of $D$ if and only if it is a semiample model of $D$ and $\\phi_* D$ is ample.", "\n\nNote that composing with an isomorphism of the target does not change the notion of ample or semiample model, so it is natural to say that two ample or semiample models $\\phi_1\\colon Z\\dasharrow Y_1$, $\\phi_2\\colon Z\\dasharrow Y_2$ are equivalent if there is an isomorphism $\\theta\\colon Y_1\\iso Y_2$ such that $\\phi_2=\\theta\\circ \\phi_1$. Then Lemma \\[lem:semiampleample\\]\\[uniqueAmple\\] says that up to equivalence, if an ample model exists then it is unique. ", "This justifies that we can speak of *the* ample model of a divisor $D$.\n\n\\[rem:Proj\\] We say that two divisors $D$ and $D'$ are *Mori equivalent* if they have the same ample model.", "\n\nFor a $\\Q$-divisor, the ample model of $D$, if it factorises through a semiample model, is the rational map $\\phi_D$ associated with the linear system $\\lvert mD\\rvert $ for $m \\gg 0$, whose image is $Z_D= \\Proj (\\bigoplus_{m} H^0(Z,mD))$, where the sum is over all positive integers $m$ such that $mD$ is Cartier (see [@KKL Remark 2.4(ii)]). ", "It does exist if the ring $\\bigoplus_{m} H^0(Z,mD)$ is finitely generated, which is for instance true if $D=K_Z+A$ for some ample $\\Q$-divisor $A$ (follows from [@BCHM Corollary 1.1.2]).", "\n\nLet $V_\\Q$ be a $\\Q$-vector space, and $V_\\R = V_\\Q \\otimes \\R$ the associated real vector space. ", "Recall that a *rational polytope* in $V_\\R$ is the convex hull of finitely many points lying in $V_\\Q$. In particular, it is convex and compact.", "\n\n\\[pro:coneC\\] Let $Z$ be a smooth variety with $K_Z$ not pseudo-effective and let $A_1,\\ldots,A_s$ be ample $\\Q$-divisors that generate the $\\R$-vector space $N^1(Z)$. Assume that there exist ample effective $\\Q$-divisors $A, A_1', \\dots, A_s'$ such that for each $i$, $A_i = A + A_i'$. Define $$\\begin{gathered}\n\\Cl= \\Bigl\\{D\\in \\Div(Z)_\\R \\Bigm| D=a_0 K_Z + \\sum_{i=1}^s a_i A_i, \\\\ a_0,\\ldots,a_s\\ge 0 \\text{ and }D \\text{ is pseudo-effective}\\Bigr\\}.\\end{gathered}$$ Then, every element of $\\Cl$ has an ample model, and the Mori equivalence classes give a finite partition $$\\Cl=\\coprod_{i\\in I} \\Al_i.$$ We denote by $\\phi_i\\colon Z\\dasharrow Z_i$ the common ample model of all $D \\in \\Al_i$. Then, the following holds $($each $i,j$ is always assumed to be in $I$ in the next statements$)$:\n\n1. ", " \\[coneC:polytope\\] The set $\\Cl$ is a cone over a rational polytope.", "\n\n2. ", " \\[coneC:Ai\\] Each $\\Al_i$ is a finite union of relative interiors of cones over rational polytopes.", "\n\n3. ", " \\[coneC:MaxDim\\] For each $i$, the following are equivalent:\n\n 1. ", " The image of $\\Al_i$ in $N_1(Z)$ has non-empty interior;\n\n 2. ", " $\\phi_i$ is birational and $Z_i$ is $\\Q$-factorial;\n\n 3. ", " $\\phi_i$ is a birational contraction that is the output of a $(K_Z+\\Delta)$-MMP for some $K_Z+\\Delta\\in \\Cl$;\n\n4. ", " \\[coneC:phi\\_iBirational\\] If $\\phi_j$ is birational, then $\\widebar{\\Al_j}$ is a cone over a rational polytope. ", "Taking a resolution $p\\colon W\\to Z$, $q\\colon W\\to Z_j$ of $\\phi_j$, we have $$\\begin{aligned}\n \\Al_j &= \\{ D\\in \\Cl \\mid \\phi_{j*} D \\text{ is ample and } p^* D -q^*\\phi_{j*}D\\ge 0\\}, \\\\\n \\widebar{\\Al_j} &=\\{ D\\in \\Cl \\mid \\phi_{j*} D \\text{ is nef and }p^* D -q^*\\phi_{j*}D\\ge 0\\}.\\end{aligned}$$ Moreover, for each $D\\in \\widebar{\\Al_j}$, the divisor $\\phi_{j*} D$ is semiample, so we also have $$\\widebar{\\Al_j}=\\{ D\\in \\Cl \\mid \\phi_j \\text{ is a semiample model of }D\\}.$$\n\n5. ", " \\[coneC:phi\\_ji\\] If $i,j$ are such that $\\widebar{\\Al_j}\\cap \\Al_i\\not=\\emptyset$, there exists a morphism $\\phi_{ji}\\colon Z_j\\to Z_i$ with connected fibres such that $\\phi_i=\\phi_{ji}\\circ \\phi_j$.\n\n 1. ", " \\[AljAli\\] If moreover $\\phi_j$ is birational, we obtain $$\\begin{aligned}\n \\widebar{\\Al_j}\\cap \\widebar{\\Al_i}&=\\{ D\\in \\widebar{\\Al_j}\\mid \\phi_{j*}D\\cdot C=0 \\text{ for each }C\\in N_1(Z_j/Z_i)\\},\\\\\n \\widebar{\\Al_j}\\cap \\Al_i&=\\{ D\\in \\widebar{\\Al_j}\\cap \\widebar{\\Al_i}\\mid \\phi_{i*}D \\text{ is ample }\\}.\\end{aligned}$$\n\n 2. ", " \\[AliAlj\\] If furthermore $\\phi_i$ is birational, we also have $$\\begin{aligned}\n \\widebar{\\Al_j}\\cap \\widebar{\\Al_i}&=\\{ D\\in \\widebar{\\Al_i}\\mid \\phi_{j*}D\\cdot C=0 \\text{ for each }C\\in N_1(Z_j/Z_i)\\}.\\end{aligned}$$\n\n6. ", " \\[coneC:singularities\\] Each variety $Z_i$ is normal and klt. ", "In particular, it has rational singularities.", "\n\n7. ", " \\[coneC:numericalonly\\] For each numerically equivalent divisors $D,D'\\in \\Cl$ and each $i$, we have $$\\begin{aligned}\n D\\in \\Al_i\\iff D'\\in \\Al_i &&\n \\text{and} &&\n D\\in \\widebar{\\Al_i}\\iff D'\\in \\widebar{\\Al_i}.\\end{aligned}$$\n\nThe finite partition, the assertions \\[coneC:Ai\\], \\[coneC:MaxDim\\], the fact that $\\widebar{\\Al_i}$ is a cone over a rational polytope if $\\phi_i$ is birational (first part of \\[coneC:phi\\_iBirational\\]) and the existence of the morphism $\\phi_{ji}$ in \\[coneC:phi\\_ji\\] are given by [@HMcK Theorem 3.3]. ", "Indeed, we can apply their result with (in their notation) the affine subspace $V \\subset \\Div(Z)_\\R$ generated by $A_1',\\ldots,A_s'$ and $-A$. Observe that they normalise their divisors by $a_0 = 1$, so they work with an affine section of our cone $\\Cl$.\n\nAs the set of pseudo-effective divisors is a closed convex cone, so is $\\Cl$. Moreover, $\\Cl$ is rational and polyhedral, as it is the union of finitely many rational polyhedral cones. ", "This gives \\[coneC:polytope\\].", "\n\nWe now prove the remaining part of \\[coneC:phi\\_iBirational\\]. ", "By Lemma \\[lem:semiampleample\\]\\[birsemiampleample\\], for each $D\\in \\Cl$, $\\phi_j$ is the ample model of $D$ if and only if it is a semiample model of $D$ and $\\phi_{j*} D$ is ample. ", "This corresponds exactly to asking that $\\phi_{j*} D$ is ample and $p^* D-q^*\\phi_* D $ is effective. ", "The closure of ample divisors being nef divisors, we obtain the explicit description of $\\Al_j $ and the first description of $\\widebar{\\Al_j}$ given in \\[coneC:phi\\_iBirational\\].", "\n\nThe fact that $\\phi_{j*} D$ is semiample for each $D\\in \\widebar{\\Al_j}$ is part of the proof (and in fact the main point) of the existence of $\\phi_{ji}\\colon Z_j\\to Z_i$ given in [@HMcK Theorem 3.3]. ", "It implies that $\\widebar{\\Al_j}=\\{ D\\in \\Cl \\mid \\phi_j \\text{ is a semiample model of }D\\}$.\n\nWe now prove the remaining part of \\[coneC:phi\\_ji\\]. ", "If $i=j$, everything follows with $\\phi_{ii}=\\id_{Z_i}$, so we can assume that $i\\not=j$.\n\nWe first prove \\[coneC:phi\\_ji\\]\\[AljAli\\]. ", "For each element $D\\in \\widebar{\\Al_j}$, we denote by $H_j$ the divisor $H_j=\\phi_{j*}D$, which is semiample by \\[coneC:phi\\_iBirational\\], and obtain $p^* D=q^*H_j+E$, where $E$ is effective and $q$-exceptional. ", "We then use the resolution of $\\phi_i$ given by $p\\colon W\\to Z$ and $q'=\\phi_{ji}\\circ q\\colon W\\to Z_i$ to determine if $D$ belongs to $\\Al_i$ or its closure.", "\n\n1. ", " Suppose first that $D\\in \\Al_i$, which means that $\\phi_i$ is the ample model of $D$. There exists then an ample divisor $H_i$ on $Z_i$ such that $p^*D$ is linearly equivalent to $(\\phi_{ji}\\circ q)^* H_i+E'$ where $E'\\ge0$ and where $R\\ge E'$ for each effective $\\R$-divisor $R$ linearly equivalent to $p^*D$. As $p^* D =q^* H_j+E$ and as $q^* H_j$ has no fixed component (because $H_j$ is semiample), we obtain $E\\ge E'$, so $E'$ is $q$-exceptional. ", "Hence, $\\phi_{j*} D$ is linearly equivalent to $q_*((\\phi_{ji}\\circ q)^* H_i+E')=\\phi_{ji}^* H_i$ and thus satisfies $\\phi_{j*} D\\cdot C=0$ for each $C\\in N_1(Z_j/Z_i)$. Moreover, $\\phi_{i*} D$ is linearly equivalent to $H_i$ and is thus ample.", "\n\n2. ", " Conversely, we suppose that $\\phi_{j*} D\\cdot C=0$ for each $C\\in N_1(Z_j/Z_i)$ and that $\\phi_{i*} D$ is ample, and prove that $D\\in \\Al_i$.\n\n We denote by $H_i$ the ample divisor $\\phi_{i*} D$. Since the nef divisor $H_j=\\phi_{j*} D$ on $Z_j$ satisfies $H_j\\cdot C=0$ for each $C\\in N_1(Z_j/Z_i)$ and satisfies $\\phi_{ji*} H_j=H_i$, the divisor $H_j-\\phi_{ji}^* H_i$ is numerically trivial. ", "We obtain $p^* D \\equiv (\\phi_{ji}\\circ q)^*H_i+E$. It remains to see that if $R$ is an effective divisor linearly equivalent to $p^* D$, then $R\\ge E$. We write $$M +F = R \\equiv (\\phi_{ji}\\circ q)^* H_i+E$$ with $|M|$, $F$ respectively the mobile and fixed part of the linear system $|R|$. The divisor $(\\phi_{ji}\\circ q)^* H_i = q^* (\\phi_{ji}^* H_i)$ is $q$-trivial and mobile, so $E - F \\ge 0$ with support in the exceptional locus of $q$. So we have $$M + F = R = q\\text{-trivial} + E \\implies E - F = q\\text{-trivial} + M$$ and we conclude by the Negativity Lemma \\[lem:Negativity\\] that $F-E \\ge 0$, hence $F-E = 0$, and $R - E = M \\ge 0$. This achieves the proof of \\[coneC:phi\\_ji\\]\\[AljAli\\].", "\n\nWe now prove \\[coneC:phi\\_ji\\]\\[AliAlj\\]. ", "The fact that $\\phi_{j*}D\\cdot C=0$ for each $C\\in N_1(Z_j/Z_i)$ and each $D\\in \\widebar{\\Al_j}\\cap \\widebar{\\Al_i}$ follows from \\[coneC:phi\\_ji\\]\\[AljAli\\]. ", "Conversely, we take $D\\in \\widebar{\\Al_i}$ that satisfies $\\phi_{j*}D\\cdot C=0$ for each $C\\in N_1(Z_j/Z_i)$, and prove that $D\\in \\widebar{\\Al_j}$, using the diagram $$\\begin{tikzcd}[link]\n& W \\ar[dl,\"p\",swap] \\ar[dr,\"q\"]\\\\\nZ \\ar[rr,\"\\phi_j\",dashed] \\ar[rrr, bend right=10, \"\\phi_i\",swap,dashed] && Z_j \\ar[r,\"\\phi_{ji}\"] & Z_i\n\\end{tikzcd}$$ As $D\\in \\widebar{\\Al_i}$, by \\[coneC:phi\\_iBirational\\] the divisor $H_i=\\phi_{i*}D$ is nef and $p^*D=q^*\\phi_{ji}^*H_i+E_i$ where $E_i\\ge 0$ is $(\\phi_{ji}\\circ q)$-exceptional. ", "Hence, $H_j=\\phi_{j*}D=q_* p^* D =\\phi_{ji}^*H_i+q_* E_i$, where $q_* E_i\\ge 0$ is $\\phi_{ji}$-exceptional. ", "As $H_j\\cdot C=0$ for each $C\\in N_1(Z_j/Z_i)$, we also obtain $q_* E_i\\cdot C=0$ for each $C\\in N_1(Z_j/Z_i)$. The Negativity Lemma \\[lem:Negativity\\] gives $q_* E_i\\le 0$, so $q_* E_i=0$. This implies that $H_j={\\phi_{ji}}^*H_i$ is nef (because $H_i$ is nef) and that $p^* D=q^*H_j+E_i$, where $E_i\\ge 0$ is $q$-exceptional, so \\[coneC:phi\\_iBirational\\] gives $D\\in \\widebar{\\Al_j}$ as expected.", "\n\nFinally we prove \\[coneC:singularities\\] and \\[coneC:numericalonly\\], for a given index $i$. The variety $Z_i$ is normal by definition of an ample model. ", "If $\\Al_i$ satisfies the equivalent conditions of \\[coneC:MaxDim\\], then $Z_i$ is terminal (hence has rational singularities) as the output of a $(K_Z+\\Delta)$-MMP, and the fact that belonging to $\\Al_i$ or $\\widebar{\\Al_i}$ is a numerical condition follows from \\[coneC:phi\\_iBirational\\]. ", "To see this, we take $D\\in \\Cl$, and write $p^* D -q^*\\phi_{j*}D=\\sum_{i=1}^l a_i E_i$ where $E_1,\\ldots,E_l$ are the divisors contracted by the birational morphism $q\\colon W\\to Z_j$. Then we observe that the real numbers $a_i$ can be computed by intersecting $E$ with linear combinations of curves on the exceptional divisors. ", "Indeed there does not exist any $(\\mu_1,\\ldots,\\mu_l)\\in \\R^l\\setminus \\{0\\}$ such that $E=\\sum_{i=1}^l \\mu_i E_i$ intersects trivially each element of $N_1(W/Z_j)$. This follows from the Negativity Lemma \\[lem:Negativity\\] applied to $E$ and $-E$.\n\nIf $\\Al_i$ does not satisfies the equivalent conditions of \\[coneC:MaxDim\\], there exists a chamber $\\Al_j$ satisfying them, such that $\\widebar{\\Al_j} \\cap \\Al_i \\neq\\emptyset$. By \\[coneC:phi\\_ji\\] there is a contraction $\\phi_{ji}\\colon Z_j\\to Z_i$ and $Z_j$ is $\\Q$-factorial and terminal again by \\[coneC:MaxDim\\]. ", "Assertion \\[coneC:singularities\\] follows from Proposition \\[pro:sing\\_of\\_Proj\\] and Remark \\[rem:Proj\\]. ", "Assertion \\[coneC:numericalonly\\] follows from \\[coneC:phi\\_ji\\].", "\n\n\\[lem:complements\\] Assume the setting of Proposition $\\ref{pro:coneC}$. Let $\\Al_j \\subseteq \\Cl$ be a Mori equivalence class such that $\\phi_j\\colon Z\\rat Z_j$ is birational.", "\n\n1. ", " \\[comp:combination\\] If $\\{D_i\\}$ is a finite collection of classes in $N^1(Z)$ such that $\\phi_j$ is a semiample model of each, then $\\phi_j$ is a semiample model for any convex combination of the $D_i$.\n\n2. ", " \\[comp:model\\] Let $D_j\\in \\Nef(Z_j)$ and set $D = \\phi_j^* D_j \\in N^1(Z)$. Then $\\phi_j$ is a semiample model of $D$.\n\n3. ", " \\[comp:exceptional\\] Let $E$ be a prime divisor contracted by $\\phi_j$. Then $\\phi_j$ is a semiample model of $E$.\n\n4. ", " \\[comp:Aj=\\] The cone $\\widebar{\\Al_j}$ is the intersection of $\\Cl$ with the closed convex cone generated by $\\phi_j^* \\Nef(Z_j)$ and by the divisors contracted by $\\phi_j$.\n\n\\[comp:combination\\]. ", "For each $i$ we write $p^* D_i = q^*H_i + E_i$ where $H_i=\\phi_{j*} D_i$ is semiample, and $E_i \\ge 0$ is $q$-exceptional. ", "For any choice of coefficients $a_i \\ge 0$ we have $$p^* \\Bigl( \\sum a_i D_i \\Bigr) = q^* \\Bigl( \\sum a_i H_i \\Bigr) + \\sum a_i E_i$$ where $ \\sum a_i H_i =\\phi_{j*}( \\sum a_i D_i)$ is semiample and $\\sum a_i E_i$ is $q$-exceptional and effective.", "\n\n\\[comp:model\\]. ", "We take $p\\colon W\\to Z$ and $q\\colon Z\\to Z_j$ a resolution of $\\phi_j$. We obtain $D=p_* q^*D_j$. The divisor $E=p^*D-q^*D_j$ is $p$-exceptional, and also $q$-exceptional since $\\phi_j$ is a birational contraction, hence $\\phi_{j*} D=q_* p^*D =D_j$. Moreover, $-E=q^*D_j-p^*D$ is $p$-nef, so $E\\ge0$ by the Negativity Lemma \\[lem:Negativity\\].", "\n\n\\[comp:exceptional\\]. ", "The trivial divisor $H=\\phi_{j*} E = 0$ is semiample, and $p^* E -q^* H = p^*E$ is effective, as $E$ is effective.", "\n\n\\[comp:Aj=\\]. ", "As $\\phi_j$ is birational, by Proposition \\[pro:coneC\\]\\[coneC:phi\\_iBirational\\] we have $$\\widebar{\\Al_j}=\\{ D\\in \\Cl \\mid \\phi_j \\text{ is a semiample model of }D\\}.$$ By \\[comp:model\\] and \\[comp:exceptional\\], $\\phi_j$ is a semiample model of every element of $\\phi_j^* \\Nef(Z_j)$ and of every prime divisor contracted by $\\phi_j$. So, by \\[comp:combination\\], $\\widebar{\\Al_j}$ contains the intersection of $\\Cl$ with the closed convex cone generated by $\\phi_j^* \\Nef(Z_j)$ and by the divisors contracted by $\\phi_j$. Conversely, let $D \\in \\widebar{\\Al_j}$, and set $D_j = \\phi_{j*}D$. The divisor $$E = D - {\\phi_j}^*D_j =p_* p^*D - p_* q^* D_j = p_*(p^*D-q^* D_j )$$ is effective and $\\phi_j$-exceptional, so $D = {\\phi_j}^*D_j + E$ is the sum of an element in $\\phi_j^* \\Nef(Z_j)$ and an effective $\\phi_j$-exceptional divisor.", "\n\n\\[setup:coneC\\] Let $Z$ be a smooth variety with $K_Z$ not pseudo-effective and let $A_1,\\ldots,A_s$ be ample $\\Q$-divisors that generate the $\\R$-vector space $N^1(Z)$. We still denote $$\\begin{gathered}\n\\Cl= \\Bigl\\{D\\in N^1(Z) \\Bigm| D=a_0 K_Z + \\sum_{i=1}^s a_i A_i, \\\\ a_0,\\ldots,a_s\\ge 0 \\text{ and }D \\text{ is pseudo-effective}\\Bigr\\}.\\end{gathered}$$ This is the image under the natural map $\\Div(Z)_\\R \\to N^1(Z)$ of the cone from Proposition \\[pro:coneC\\], for some choice of ample effective $\\Q$-divisors $A, A_1', \\dots, A_s'$ such that for each $i$, $A_i \\equiv A + A_i'$. By Proposition \\[pro:coneC\\]\\[coneC:numericalonly\\], the decomposition $\\Cl = \\coprod_{i\\in I} \\Al_i$ (hence also its image in $N^1(Z)$) does not depend on such a choice of effective representatives. ", "So from now on we will work directly in the finite dimensional $\\R$-vector space $N^1(Z)$, and use the notation $\\Cl, \\Al_i$ in this context only.", "\n\n\\[rem:klt\\] One advantage of working up to numerical equivalence is that we can always assume that the pairs $(Z,\\Delta)$ in Set-Up \\[setup:coneC\\] are klt with arbitrary small discrepancies, where $\\Delta = \\frac{1}{a_0} \\sum_{i=1}^s a_i A_i$. Indeed, by expressing each $A_i$ as $A_i \\equiv \\frac1N \\sum_{j = 1}^N H_{i,j}$ for some large integer $N$ and some general members $H_{i,j} \\in \\lvert A_i\\rvert$, we can ensure that the union of the supports of the $H_{i,j}$ is a simply normal crossing divisor and that all coefficients appearing in the convex combination $\\Delta$ are positive and very small.", "\n\nAssuming Set-Up \\[setup:coneC\\], we introduce some terminology. ", "We say that a chamber $\\Al_i$ has *maximal dimension* if it has non-empty interior in $N^1(Z)$, which corresponds to the equivalent assertions of Proposition \\[pro:coneC\\]\\[coneC:MaxDim\\]. ", "We say that a chamber $\\Al_i$ is *big* if all divisors (or equivalently, one divisor) in $\\Al_i$ are big. ", "We call *face* of $\\Cl$ any face of a polyhedral cone $\\widebar{\\Al_i} \\subseteq \\Cl \\subseteq N^1(Z)$, for any big chamber $\\Al_i$. By the *codimension* of a face in $\\Cl$ we always mean the codimension in $N^1(Z)$ of the smallest vector subspace containing it. ", "We will usually denote $\\Fl^r$ a face of codimension $r$ in $\\Cl$, and $\\interior{\\Fl}^r$ its relative interior.", "\n\nWe denote by $\\partial^+\\Cl$ the set of non-big divisors in $\\Cl$. As $\\partial^+\\Cl$ is the intersection of $\\Cl$ with the boundary of the pseudo-effective cone, the set $\\partial^+\\Cl$ is a closed subset of the boundary of $\\Cl$. We have $\\Al_i\\subseteq \\partial^+\\Cl$ if $\\dim Z_i<Z$ and $\\Al_i\\subseteq \\Cl\\setminus \\partial^+\\Cl$ if $\\dim Z_i=Z$.\n\nBy definition, the cone $\\Cl \\subset N^1(Z)$ is equal to the intersection of two convex closed cones, namely $\\Cl = \\Cl' \\cap \\widebar{\\Eff}(Z)$ with $\\Cl'$ the convex cone generated by $K_Z$ and the $A_i$. We will say that a face $\\Fl \\subseteq \\Cl$ is *inner* if it meets the interior of $\\Cl'$. In particular, $\\Fl$ is inner if for any $D' \\in \\interior{\\Fl}$, there exists a neighborhood $V$ of $D'$ in $N^1(Z)$ such that $\\widebar{\\Eff}(Z) \\cap V = \\Cl \\cap V$. Equivalently, a face is inner if it meets either the interior of $\\Cl$ or the relative interior of $\\partial^+ \\Cl$.\n\n\\[rem:InnerFace\\] If $\\Fl$ is an inner face, then for any $D \\in \\widebar{\\Eff}(Z)$ and any $D'\\in \\interior{\\Fl}$, we have $D'+\\eps D \\in \\Cl$ for sufficiently small $\\eps \\ge 0$. Indeed with the notation above one can choose $V \\subset \\Cl'$ a neighborhood of $D'$ such that $\\widebar{\\Eff}(Z) \\cap V = \\Cl \\cap V$. Then it suffices to choose $\\eps$ such that $D'+\\eps D \\in V$. Since $D, D'$ are both pseudo-effective, the segment $[D,D']$ also is contained in the convex cone $\\widebar{\\Eff}(Z)$, and the claims follows.", "\n\n\\[lem:faces\\_factory\\] Let $\\Al_j$ be a big inner chamber.", "\n\n1. ", " \\[faces:Fji\\] Let $\\Al_i$ be a chamber such that $\\widebar{\\Al_j} \\cap \\Al_i \\neq \\emptyset$. Then $\\Fl_{ji} := \\widebar{\\Al_j} \\cap \\widebar{\\Al_i}$ has the following properties:\n\n 1. ", " \\[FjiFaceAj\\] $\\Fl_{ji}$ is a face of $\\widebar{\\Al_j}$.\n\n 2. ", " \\[FjiFaceAi\\] If $\\Al_i$ is a big inner chamber, then $\\Fl_{ji}$ is also a face of $\\widebar{\\Al_i}$.\n\n 3. ", " \\[faces:EinFji\\] If $\\widebar{\\Al_j}$ has maximal dimension, then the vector space $$V_{ji}:=\\{ D\\in N^1(Z)\\mid \\phi_{j*}D\\cdot C=0 \\text{ for each }C\\in N_1(Z_j/Z_i)\\}$$ is spanned by $\\Ex(\\phi_j)$ and $\\phi_i^*\\Nef(Z_i)$, and has codimension $\\rho(Z_j/Z_i)$ in $N^1(Z)$. Moreover if $\\Fl_{ji}$ is inner, then $V_{ji}$ is spanned by $\\Fl_{ji}$.\n\n 4. ", " \\[faces:Fji\\_in\\_Ai\\] $\\interior{\\Fl}_{ji} \\subseteq \\Al_i$.\n\n2. ", " \\[faces:Fjk\\_in\\_Fji\\] Let $\\Al_i$, $\\Al_k$ be two chambers such that $\\widebar{\\Al_j} \\cap \\Al_i \\neq \\emptyset$, $\\widebar{\\Al_j} \\cap \\Al_k \\neq \\emptyset$ and $\\widebar{\\Al_i} \\cap \\Al_k \\neq \\emptyset$. Then the face $\\Fl_{jk}$ is a subface of the face $\\Fl_{ji}$, and a strict subface if $i \\neq k$.\n\n3. ", " \\[faces:inner\\_is\\_inter\\] Conversely, any inner face $\\Fl^r \\subseteq \\Cl$ is of the form $\\Fl^r= \\Fl_{ki} := \\widebar{\\Al_k}\\cap\\widebar{\\Al_i}$ for some chamber $\\Al_k$ of maximal dimension, and some chamber $\\Al_i$ containing $\\interior{\\Fl}^r$.\n\nWe first prove Assertions \\[FjiFaceAj\\]-\\[faces:EinFji\\] of \\[faces:Fji\\]. ", "By Proposition \\[pro:coneC\\]\\[coneC:phi\\_ji\\]\\[AljAli\\], $$\\widebar{\\Al_j}\\cap \\widebar{\\Al_i} =\\{ D\\in \\widebar{\\Al_j}\\mid \\phi_{j*}D\\cdot C=0 \\text{ for each } C\\in N_1(Z_j/Z_i)\\}.$$ So each curve $C \\in N_1(Z_j/Z_i)$ defines a supporting hyperplane of $\\widebar{\\Al_j}$, and $\\widebar{\\Al_j}\\cap \\widebar{\\Al_i}$ is a face of $\\widebar{\\Al_j}$ as the intersection of a cone over a polytope with a collection of supporting hyperplanes. ", "This proves \\[FjiFaceAj\\]. ", "The same argument using Proposition \\[pro:coneC\\]\\[coneC:phi\\_ji\\]\\[AliAlj\\] gives \\[FjiFaceAi\\].", "\n\nWe now prove \\[faces:EinFji\\]. ", "We denote by $E_j, V_j,V_i\\subseteq N^1(Z)$ the vector spaces generated by $\\Ex(\\phi_j)$, $\\phi_j^*\\Nef(Z_j)$ and $\\phi_i^*\\Nef(Z_i)$ respectively. ", "As $\\widebar{\\Al_j}$ has maximal dimension, $\\phi_j$ is a birational contraction that is the output of a $(K_Z+\\Delta)$-MMP for some $K_Z+\\Delta\\in \\Cl$ (Proposition \\[pro:coneC\\]\\[coneC:MaxDim\\]). ", "This gives $N^1(Z)=E_j\\oplus V_j$. Now, $\\phi_j^* \\Nef(Z_j)$ contains $\\phi_i^* \\Nef(Z_i) = \\phi_j^* \\phi_{ji}^* \\Nef(Z_i)$, and the codimension of $\\phi_i^* \\Nef(Z_i)$ in $\\phi_j^* \\Nef(Z_j)$ is the same as the codimension of $\\phi_{ji}^* \\Nef(Z_i)$ in $\\Nef(Z_j)$ (indeed, the group homomorphism $\\phi_j^*\\colon N_1(Z_j)\\to N_1(Z)$ is injective as $\\phi_j$ is a birational contraction). ", "We obtain $\\dim(V_j)-\\dim(V_i)=\\rho(Z_j/Z_i)$. The vector space $V_{ji}$ is of codimension $\\dim N_1(Z_j/Z_i)=\\rho(Z_j/Z_i)$ since $\\phi_j$ is birational. ", "Since $E_j\\oplus V_i\\subseteq V_{ji}$ and both vector space have the same dimension, we obtain $$V_{ji}=E_j\\oplus V_i.$$ It remains to prove the last sentence of \\[faces:EinFji\\]. ", "To do this, we assume moreover that $\\Fl_{ji}$ is inner, and show that $\\Fl_{ji}$ spans $V_{ji}$ in this case. ", "Writing $$S_j=\\{D\\in N^1(Z)\\mid \\phi_j \\text{ is a semiample model of }D\\},$$ Proposition \\[pro:coneC\\]\\[coneC:phi\\_iBirational\\]&\\[coneC:phi\\_ji\\]\\[AljAli\\] yields $$\\widebar{\\Al_j}= \\Cl\\cap S_j,\\quad \\widebar{\\Al_j}\\cap \\widebar{\\Al_i}=\\Fl_{ji}=V_{ji}\\cap \\Cl \\cap S_j.$$ In particular, we have $\\Fl_{ji}\\subseteq V_{ji}$. To show that $\\Fl_{ji}$ spans $V_{ji}=E_j\\oplus V_i$, it suffices to prove that every divisor $E$ which is either an exceptional divisor of $\\phi_j$ or an element of $\\phi_i^*\\Nef(Z_i)$ lies in $\\mathrm{span}(\\Fl_{ji})$. We fix some $D' \\in \\interior{\\Fl}_{ji}$ and may assume that $D'+E\\in \\Cl$; indeed, we may replace $E$ with $\\eps E$ for some small $\\eps>0$, and then the result follows from $E\\in \\widebar{\\Eff}(Z)$ and Remark \\[rem:InnerFace\\]. ", "This implies that $E\\in \\widebar{\\Al_j}$ (Lemma \\[lem:complements\\]\\[comp:Aj=\\]), and thus that $E\\in \\Fl_{ji}=V_{ji}\\cap \\Cl \\cap S_j$, since $\\widebar{\\Al_j}=\\Cl\\cap S_j$ and $E\\in V_{ji}$.\n\n\\[faces:Fjk\\_in\\_Fji\\]. ", "By Proposition $\\ref{pro:coneC}\\ref{coneC:phi_ji}$ we have a commutative diagram $$\\begin{tikzcd}[link]\nZ_j \\ar[rr,\"\\phi_{jk}\"] \\ar[dr,\"\\phi_{ji}\",swap] && Z_k \\\\\n& Z_i \\ar[ur,\"\\phi_{ik}\",swap]\n\\end{tikzcd}$$ Any curve contracted by $\\phi_{ji}$ is also contracted by $\\phi_{jk}$, so the collection of supporting hyperplanes defining the face $\\Fl_{ji}$ is a subcollection of the one defining $\\Fl_{jk}$ and moreover a strict subcollection if $N_1(Z_j/Z_i) \\subsetneq N_1(Z_j/Z_k)$.\n\nWe can now prove \\[faces:Fji\\]\\[faces:Fji\\_in\\_Ai\\]. ", "Let $D \\in \\Fl_{ji}$, and assume that $D \\in \\Al_k$ with $k \\neq i$. By \\[faces:Fjk\\_in\\_Fji\\], the divisor $D$ is in $\\Fl_{jk}$ which is a strict subface of $\\Fl_{ji}$, so $D \\not\\in \\interior{\\Fl}_{ji}$. This proves $ \\interior{\\Fl}_{ji}\\subseteq \\Al_i$.\n\n\\[faces:inner\\_is\\_inter\\]. ", "If $r=0$, then $\\Fl^r$ is equal to $\\widebar{\\Al_i}$ for some chamber $i$ of maximal dimension, so we simply choose $i=k$. Now assume that $\\Fl^r$ is an inner face of codimension $r \\ge 1$. By definition $\\Fl^r$ is a face of $\\widebar{\\Al_k}$ for some big chamber $\\Al_k$. Among all possible choices for such a chamber $\\Al_k$ we pick one such that $\\rho(Z_k)$ is maximal. ", "Let $D \\in \\interior{\\Fl}^r$, and $\\Al_i$ the chamber containing $D$. If $D$ is not big, then $i \\neq k$. If $D$ is big, then since $\\Fl^r$ is inner there exists at least one big chamber $\\Al_{l}$ distinct from $\\Al_i$ such that $D \\in \\widebar{\\Al_l} \\cap \\Al_i$. This implies that $\\Fl^r \\subseteq \\Fl_{li}$ is a face of $\\widebar{\\Al_l}$, and by Proposition \\[pro:coneC\\]\\[coneC:phi\\_ji\\] we have a morphism $Z_l \\to Z_i$. Since $l \\neq i$ we get $\\rho(Z_l) > \\rho(Z_i)$, and the maximality of $\\rho(Z_k)$ ensures that $i \\neq k$. By \\[faces:Fji\\] and \\[faces:Fji\\_in\\_Ai\\] we get $\\Fl^r \\subseteq \\Fl_{ki}$. If $r = 1$, this gives $\\Fl^r = \\Fl_{ki}$.\n\nFinally we consider the case where $r \\ge 2$, and prove by contradiction that the inclusion $\\Fl^r\\subseteq \\Fl_{ki}$ is an equality. ", "If $\\Fl^r\\subsetneq \\Fl_{ki}$ is a strict subface, then there exists an inner codimension 1 face $\\Fl^1$ of $\\widebar{\\Al_k}$ such that $\\Fl^r \\subseteq \\Fl^1 \\cap \\Fl_{ki} \\subsetneq \\Fl_{ki}$. By the first part of the proof, there exists a chamber $\\Al_l$ such that either $\\Fl^1 = \\Fl_{kl}$ or $\\Fl^1 = \\Fl_{lk}$. But in the latter case, we would have $\\widebar{\\Al_l} \\cap \\Al_k \\neq\\emptyset$, hence a morphism $\\phi_{lk}\\colon Z_l \\to Z_k$, contradicting the maximality of $\\rho(Z_k)$. So $\\Fl^1 = \\Fl_{kl}$, so we have $\\widebar{\\Al_k} \\cap \\Al_l \\neq \\emptyset$, and moreover since $D \\in \\widebar{\\Al_k} \\cap \\widebar{\\Al_l} \\cap \\Al_i$, we also have $\\widebar{\\Al_k} \\cap \\Al_i \\neq \\emptyset$ and $\\widebar{\\Al_l} \\cap \\Al_i \\neq \\emptyset$. By \\[faces:Fjk\\_in\\_Fji\\], $\\Fl_{ki}$ is a subface of $\\Fl_{kl}$, so we get $\\Fl_{kl} \\cap \\Fl_{ki} = \\Fl_{ki}$, which contradicts $ \\Fl^1 \\cap \\Fl_{ki} \\subsetneq \\Fl_{ki}$, since $\\Fl_{kl}=\\Fl^1$.\n\n\\[not:Fij\\] Lemma \\[lem:faces\\_factory\\] provides the following indexing system for faces. ", "Any inner face can be written $\\Fl_{ji} := \\widebar{\\Al_j} \\cap \\widebar{\\Al_i}$, for some chamber $\\Al_j$ of maximal dimension and some chamber $\\Al_i$ such that $\\interior{\\Fl}_{ji} \\subseteq \\Al_i$. The index $i$ is uniquely defined by this last property, but there might be several possible choices for the index $j$. For instance, if we have a log-flip from $Z_j$ to $Z_k$, over a non $\\Q$-factorial $Z_i$, we have $\\Fl_{ji} = \\Fl_{ki}$.\n\n\\[ex:P2blownup\\] We illustrate the definition of Mori chambers and faces on the simple example of the blow-up $Z \\to \\p^2$ at two distinct points $p_1$ and $p_2$. Using the notation above, there are eight Mori chambers $\\Al_0,\\ldots,\\Al_7$, corresponding to morphisms $\\phi_i\\colon Z\\to Z_i$, $i=0,\\ldots,7$ to the varieties $Z_0=Z$, $Z_1=Z_2=\\F_1$, $Z_3=\\F_0$, $Z_4=\\p^2$, $Z_5=Z_6=\\p^1$ and $Z_7=\\pt$ in the commutative diagram on Figure \\[fig:faces\\] ($\\phi_0$ being the identity). ", "The two morphisms $\\phi_{14},\\phi_{24}\\colon \\F_1\\to \\p^2$ are the blow-ups of $p_1,p_2\\in \\p^2$ respectively, and $\\phi_1,\\phi_2\\colon Z\\to \\F_1$ are the blow-ups of the images of $p_1$ and $p_2$. The morphisms $\\phi_{15},\\phi_{26}\\colon \\F_1\\to \\p^1$ correspond to the $\\p^1$-bundle of $\\F_1$ and $\\phi_3=\\phi_5\\times \\phi_6\\colon Z\\to\\F_0= \\p^1\\times \\p^1$.\n\nWe give the detail of the relation between these Mori chambers and the faces of the cone $\\Cl$ in Figure \\[fig:faces\\]. ", "We denote by $E_1, E_2\\subset Z$ the curves contracted onto $p_1,p_2\\in \\p^2$ respectively, by $L$ the strict transform of the line through $p_1$ and $p_2$, and by $H=L+E_1+E_1$ the pull-back of a general line. ", "The cone $\\Eff(Z)$ is the closed convex cone generated by $E_1, E_2$ and $L$, which are the only $(-1)$-curves on $Z$, while the cone $\\Nef(Z)$ is the closed convex cone generated by $H, H - E_1$ and $H-E_2$. The anti-canonical divisor $-K_Z = 3H - E_1 -E_2 = 3L +2E_1 +2E_2$ is ample. ", "In the figure we represent an affine section of the cone, and all divisors must be understood up to rescaling by an adequate homothety: for instance this is really $-\\frac{1}{7}K_Z$ that is in the same affine section as $E_1, E_2$ and $L$, but for simplicity we write $-K_Z$.\n\nThe faces $\\Fl^0_i=\\widebar{\\Al_i}$, $i=0,\\ldots,4$ are the faces of maximal dimension, the faces $\\Fl_{ji}$ (written $\\Fl^r_{ji}$ where $r$ is the codimension as usual) are as above $\\Fl_{ji}=\\widebar{\\Al_j} \\cap \\widebar{\\Al_i}$. Every face of $\\Cl=\\Eff(Z)$ is inner.", "\n\nWe can notice that the ample chamber $\\Al_0$ is the only open one and that $\\Al_7$ is the only closed one. ", "Moreover, as a hint that the behaviour of non maximal Mori chambers can be quite erratic, observe that $\\Al_7=\\widebar{\\Al_7}$ is not connected, and that neither $\\widebar{\\Al_5}$ nor $\\widebar{\\Al_6}$ is a single face.", "\n\nThis example will be continued in Example \\[ex:facesAndFib\\] below.", "\n\n$$\\begin{aligned}\n\\begin{tikzcd}[ampersand replacement=\\&,scale=.5]\n\\&|[alias=Z]| Z_0=Z \\ar[to=F12,\"\\phi_2\"] \\ar[to=F11,\"\\phi_1\",swap] \\ar[to=F0,\"\\phi_3\"] \\\\\n|[alias=F11]| Z_1=\\F_{1}\n\\& |[alias=F0]| Z_3=\\F_0 \\ar[to=P12,\"\\phi_{36}\",pos=.2,swap] \\ar[to=P11,\"\\phi_{35}\",pos=.2] \n\\&|[alias=F12]| Z_2=\\F_{1} \\ar[to=P12,\"\\phi_{26}\"] \\\\ \n|[alias=P11]| Z_5=\\p^1 \\ar[to=pt,\"\\phi_{57}\",swap] \\ar[from=F11, crossing over,\"\\phi_{15}\",swap] \n\\& |[alias=P2]| Z_4=\\p^2 \\ar[from=F11,crossing over,\"\\phi_{14}\",pos=.2] \\ar[from=F12,crossing over,\"\\phi_{24}\",pos=.2,swap] \\ar[to=pt,\"\\phi_{47}\"] \n\\& |[alias=P12]| Z_6=\\p^1 \\ar[to=pt,crossing over,\"\\phi_{67}\"] \\\\\n\\& |[alias=pt]| Z_7=\\pt\n\\end{tikzcd}\n&&\n\\begin{array}{rcl}\n\\Al_0 &=& \\interior{\\Fl}^0_0. \\\\", "\n\\Al_1 &=& \\interior{\\Fl}^0_1 \\cup \\interior{\\Fl}^1_{01}. \\\\", "\n\\Al_2&=& \\interior{\\Fl}^0_2 \\cup \\interior{\\Fl}^1_{02}. \\\\", "\n\\Al_3&=& \\interior{\\Fl}^0_3 \\cup \\interior{\\Fl}^1_{03}.\\\\\n\\Al_4&=& \\interior{\\Fl}^0_4 \\cup \\interior{\\Fl}^1_{14} \\cup \\interior{\\Fl}^1_{24} \\cup \\interior{\\Fl}^2_{04}\\\\ \n\\Al_5&=& \\interior{\\Fl}^1_{15} \\cup \\interior{\\Fl}^1_{35} \\cup \\interior{\\Fl}^2_{05}. \\\\", "\n\\Al_6&=& \\interior{\\Fl}^1_{26} \\cup \\interior{\\Fl}^1_{36} \\cup \\interior{\\Fl}^2_{06}. \\\\", "\n\\Al_7&=& \\interior{\\Fl}^1_{47} \\cup \\interior{\\Fl}^2_{17} \\cup \\interior{\\Fl}^2_{27}\\cup \\interior{\\Fl}^2_{37}.\\\\ \n\\end{array}\\end{aligned}$$\n\n$$\\begin{aligned}\n\\begin{tikzpicture}[scale=4.6,font=\\footnotesize,baseline=(-K)]\n\\coordinate (E1) at (0:0) {};\n\\coordinate (E2) at (0:1) {};\n\\coordinate (P) at (60:1) {};\n\\coordinate (H) at (barycentric cs:E1=1,E2=1,P=1) {};\n\\coordinate (P1) at (barycentric cs:E1=0,E2=1,P=1) {};\n\\coordinate (P2) at (barycentric cs:E1=1,E2=0,P=1) {};\n\\coordinate (-K) at (barycentric cs:E1=2,E2=2,P=3) {};\n\\draw (E1) to (P2)\n (P2) to (P)\n (P) to (P1)\n (P1) to (E2)\n (E2) to (E1);\n\\draw (E1) to (H)\n (E2) to (H)\n (H) to (P2)\n (P1) to (H)\n (P1) to (P2);\n\\node at (H) {$\\bullet$};\n\\node at (P1) {$\\bullet$};\n\\node at (P2) {$\\bullet$};\n\\node at (E1) {$\\bullet$};\n\\node at (E2) {$\\bullet$};\n\\node at (P) {$\\bullet$};\n\\node at (-K) {$\\bullet$};\n\\node[right,xshift=-8,yshift=3] at (-K) {$-K_Z$};\n\\node[left] at (E1) {$E_2$};\n\\node[left,yshift=4] at (P2) {$L+E_2$};\n\\node[left,yshift=-4,xshift=-2] at (P2) {$=H-E_1$};\n\\node[above] at (P) {$L$};\n\\node[right,yshift=4] at (P1) {$L+E_1$};\n\\node[right,yshift=-4,xshift=2] at (P1) {$=H-E_2$};\n\\node[right] at (E2) {$E_1$};\n\\node[below] at (H) {$H$};\n\\end{tikzpicture}\n&&\n\\begin{tikzpicture}[scale=4.6,font=\\footnotesize,baseline=(-K)]\n\\coordinate (E1) at (0:0) {};\n\\coordinate (E2) at (0:1) {};\n\\coordinate (P) at (60:1) {};\n\\coordinate (H) at (barycentric cs:E1=1,E2=1,P=1) {};\n\\coordinate (P1) at (barycentric cs:E1=0,E2=1,P=1) {};\n\\coordinate (P2) at (barycentric cs:E1=1,E2=0,P=1) {};\n \\coordinate (-K) at (barycentric cs:E1=2,E2=2,P=3) {};\n\\draw (E1) to [\"\\scriptsize $\\Fl^1_{15}$\",xshift=1] (P2)\n (P2) to [\"\\scriptsize $\\Fl^1_{35}$\",xshift=1] (P)\n (P) to [\"\\scriptsize $\\Fl^1_{36}$\",xshift=-1] (P1)\n (P1) to [\"\\scriptsize $\\Fl^1_{26}$\",xshift=-1] (E2)\n (E2) to [\"\\scriptsize $\\Fl^1_{47}$\",yshift =0.3mm] (E1);\n\\draw (E1) to [swap,\"\\scriptsize $\\Fl^1_{14}$\",yshift =1mm,xshift=-0.5](H)\n (E2) to [\"\\scriptsize $\\Fl^1_{24}$\",yshift =1mm,xshift=0.5](H)\n (H) to [\"\\scriptsize $\\Fl^1_{01}$\",yshift =1mm,xshift=0.5] (P2)\n (P1) to [\"\\scriptsize $\\Fl^1_{02}$\",yshift =1mm,xshift=-0.5] (H)\n (P1) to [\"\\scriptsize $\\Fl^1_{03}$\",swap,yshift=-.5] (P2);\n\\node at (H) {$\\bullet$};\n\\node at (P1) {$\\bullet$};\n\\node at (P2) {$\\bullet$};\n\\node at (E1) {$\\bullet$};\n\\node at (E2) {$\\bullet$};\n\\node at (P) {$\\bullet$};\n\\node[left] at (E1) {\\scriptsize $\\Fl^2_{17}$};\n\\node[left] at (P2) {\\scriptsize $\\Fl^2_{05}$};\n\\node[above] at (P) {\\scriptsize $\\Fl^2_{37}$};\n\\node[right] at (P1) {\\scriptsize $\\Fl^2_{06}$};\n\\node[right] at (E2) {\\scriptsize $\\Fl^2_{27}$};\n\\node[below,yshift=1] at (H) {\\scriptsize $\\Fl^2_{04}$};\n\\node[below,yshift=-18] at (H) {\\scriptsize $\\Fl^0_4\\!=\\!\\widebar{\\Al_4}$};\n\\node[below left,xshift=-10,yshift=3] at (H) {\\scriptsize $\\Fl^0_1\\!=\\!\\widebar{\\Al_1}$};\n\\node[above,yshift=20] at (-K) {\\scriptsize $\\Fl^0_3\\!=\\!\\widebar{\\Al_3}$};\n\\node[below right,xshift=10,yshift=3] at (H) {\\scriptsize $\\Fl^0_2\\!=\\!\\widebar{\\Al_2}$};\n\\node[above,yshift=5] at (H) {\\scriptsize $\\Fl^0_0\\!=\\!\\widebar{\\Al_0}$};\n\\end{tikzpicture}\\end{aligned}$$\n\nAs a warm-up before the next section, we let the reader check that Proposition \\[pro:coneC\\] implies the following facts about codimension $1$ faces of $\\Cl$.\n\n\\[rem:codim1\\] Let $\\Fl^1$ be an inner codimension $1$ face of the cone $\\Cl \\subseteq N^1(Z)$ from Set-Up $\\ref{setup:coneC}$, and $\\Al_i$ the Mori chamber containing $\\interior{\\Fl}^1$ given by Lemma \\[lem:faces\\_factory\\]\\[faces:Fji\\_in\\_Ai\\]. ", "Then $\\Fl^1$ is contained in the closure of exactly one or two chambers of maximal dimension, depending whether $\\Fl^1$ is in the boundary of $\\Cl$ or not.", "\n\n1. ", " Assume first that $\\Fl^1 \\subset \\widebar{\\Al_j}$ for a unique chamber $\\Al_j$ of maximal dimension, so $\\Fl^1$ is in the boundary of $\\Cl$. Moreover since $\\Fl^1$ is inner we have $\\Fl^1 \\subseteq \\partial^+ \\Cl$, so $\\dim Z_i < \\dim Z_j$. The associated map $\\phi_{ji} \\colon Z_j \\to Z_i$ satisfies $\\rho(Z_j/Z_i) = 1$. Moreover $-K_{Z_j}$ is relatively ample, so that $Z_j/Z_i$ is a Mori fibre space ([@Kaloghiros Lemma 3.2], see also Proposition \\[pro:boundary\\] below for a generalisation).", "\n\n2. ", " Now consider the case where $\\Fl^1 = \\widebar{\\Al_j} \\cap \\widebar{\\Al_k}$ for some distinct chambers $\\Al_j, \\Al_k$ of maximal dimension. ", "We distinguish two subcases.", "\n\n 1. ", " If $\\Al_i$ is of maximal dimension, up to renumbering we can assume $\\Al_i = \\Al_k$, so that $\\widebar{\\Al_j} \\cap \\Al_i \\supseteq \\interior{\\Fl}^1$. In this situation both $Z_j$ and $Z_i$ are $\\Q$-factorial and terminal, so the morphism $\\phi_{ji} \\colon Z_j \\to Z_i$ with relative Picard rank $1$ given by Proposition \\[pro:coneC\\] is a birational contraction.", "\n\n 2. ", " Finally if $\\Al_i$ is not of maximal dimension, both birational morphisms $\\phi_{ji}$ and $\\phi_{ki}$ given by Proposition \\[pro:coneC\\] have relative Picard $1$ and target variety $Z_i$ which is not $\\Q$-factorial, so $\\phi_{ji}$ and $\\phi_{ki}$ are small contractions. ", "By uniqueness of the log-flip, the induced birational map $Z_j \\ps Z_k$ must be the associated log-flip.", "\n\n\\[rem:MMPscaling\\] Let $\\Delta \\in \\Cl$ be an ample divisor. ", "Then the successive chambers of maximal dimension that are cut by the segment $[\\Delta, K_Z]$ can be interpreted as successive steps in a $K_Z$-MMP from $Z$. In [@BCHM Remark 3.10.10] this is called a $K_Z$-MMP *with scaling of $\\Delta$*. ", "Moreover by perturbing $\\Delta$ we can assume that the segment is transverse to the polyhedral decomposition. ", "Then as mentioned in Remark \\[rem:codim1\\], each intermediate face of codimension 1 that the segment meets corresponds either to a flip or to a divisorial contraction, and the last codimension 1 face in the boundary of the pseudo-effective cone corresponds to a Mori fibre space structure on the output of the MMP.", "\n\n\\[rem:references\\] In this section we chose to follow [@HMcK] and [@BCHM], who build on [@Shokurov §6] for the construction of polyhedral chambers. ", "In fact it seems to be Shokurov who coined the terminology “geography of ample models” to refer to this polyhedral decomposition.", "\n\nHowever we should point out that another possible thread of references would be to use [@CasciniLazic] and [@CortiLazic], on which rely both [@Kaloghiros] and [@KKL]. ", "In these references the polyhedral decomposition is derived from [@Ein_al], and is equivalent to our decomposition into faces.", "\n\n\\[rem:fan\\] Recall that a *fan* is a collection of rational strongly convex polyhedral cones, such that each face (of any dimension) of a cone is also part of the collection, and such that the intersection of two cones is a face of each.", "\n\nLemma \\[lem:faces\\_factory\\] almost gives a fan structure on $\\Cl$: the only missing point would be to check the property for non inner faces. ", "We do not complete this study since we will not need it, but we let the interested reader check that the collection of all faces of big chambers indeed forms a fan, which coincides with the fan structure of [@Ein_al Theorem 4.1] and [@KKL Theorem 3.2].", "\n\nGeneration and relations in the Sarkisov programme {#sec:kaloghiros}\n--------------------------------------------------\n\nThe goal of this section is to prove Theorem \\[thm:sarkisov\\], which will allow us to define the group homomorphisms of the main theorems. ", "The main technical intermediate step is Proposition \\[pro:boundary\\], which explains the relation between our notion of rank $r$ fibration and the combinatorics of the non-big boundary of the cone $\\Cl$ as given in [@Kaloghiros].", "\n\nThe proof of the following lemma can be extracted from [@HMcK Lemma 4.1].", "\n\n\\[lem:ample models\\] Let $Z$ be a smooth variety, $\\pi\\colon Z \\to X$ a birational morphism, and $\\eta \\colon X \\to B$ a terminal Mori fibre space. ", "There exist ample divisors $\\Delta, \\Delta'$ on $Z$ such that $\\pi$ is the ample model of $K_Z+\\Delta$ and the semiample model of $K_Z+\\Delta'$ and $\\eta \\circ \\pi$ is the ample model of $K_Z+\\Delta'$.\n\nAs a consequence, we get the following fact, which is essentially [@Kaloghiros Proposition 3.1(ii)].", "\n\n\\[pro:ResolutionZ\\] Let $t\\ge 2$ be an integer. ", "For $i=1,\\ldots,t$, let $\\eta_i\\colon X_i\\to B_i$ be a terminal Mori fibre space and let $\\theta_i\\colon X_i\\dasharrow X_{i+1}$ be a birational map $($here $\\theta_t$ goes from $X_t$ to $X_{t+1}:=X_1)$. We assume moreover that $\\theta_t\\circ \\cdots\\circ \\theta_1=\\id_{X_1}$.\n\nThere exists a smooth variety $Z$, together with birational morphisms $\\pi_i\\colon Z\\to X_i$, $i=1,\\ldots,t$, and ample $\\Q$-divisors $A_1,\\ldots,A_{m}$ on $Z$ such that the following hold:\n\n1. ", " \\[CommonResZgen\\] The divisors $A_1,\\ldots,A_m$ generate the $\\R$-vector space $N^1(Z)$.\n\n2. ", " \\[CommonResZisthere\\] For $i=1,\\ldots,t$, the birational morphism $\\pi_i$ and the morphism $\\eta_i\\circ\\pi_i$ are ample models of an element of $$\\Cl= \\Bigl\\{a_0 K_Z + \\sum_{i=1}^m a_i A_i \\Bigm| a_0,\\ldots,a_m\\ge 0\\Bigr\\}\\cap \\widebar{\\Eff}(Z).$$\n\n3. ", " \\[CommonResZcommut\\] For $i=1,\\ldots,t$ we have $\\theta_i\\circ \\pi_i=\\pi_{i+1}$ $($with $\\pi_{t+1}:=\\pi_1)$. We then have a commutative diagram as in Figure $\\ref{fig:CommonZ}$.\n\n$$\\begin{tikzpicture}[scale=1.1,font=\\small]\n\\node (B) at (0:0cm) {$Z$};\n\\foreach \\x in {1,...,5}{\n\\node (X\\x) at (\\x*45+45:1.8*\\Rc) {$X_{\\x}$};\n\\draw[->] (B) to [auto,\"$\\pi_\\x$\",outer sep=-2pt,pos=.6] (X\\x);\n}\n\\node (Xt) at (360+45:1.8*\\Rc) {$X_{t}$};\n\\draw[->] (B) to [auto,\"$\\pi_t$\",outer sep=-2pt,pos=.6] (Xt);\n\\node (Xtm) at (315+45:1.8*\\Rc) {$X_{t-1}$};\n\\draw[->] (B) to [auto,\"$\\pi_{t-1}$\",outer sep=-2pt,pos=.6] (Xtm);\n\\draw[dashed,->] (X1) to [bend right=10,pos=.2,swap,\"$\\theta_1$\"] (X2);\n\\draw[dashed,->] (X2) to [bend right=10,pos=.5,swap,\"$\\theta_2$\"] (X3);\n\\draw[dashed,->] (X3) to [bend right=10,pos=.4,swap,\"$\\theta_3$\"] (X4);\n\\draw[dashed,->] (X4) to [bend right=10,pos=.8,swap,\"$\\theta_4$\"] (X5);\n\\node at (-45:1.6*\\Rc) {\\reflectbox{$\\ddots$}};\n\\draw[dashed,->] (Xtm) to [bend right=10,auto,swap,\"$\\theta_{t-1}$\",pos=.3] (Xt);\n\\draw[dashed,->] (Xt) to [bend right=10,auto,swap,\"$\\theta_t$\"] (X1);\n\\end{tikzpicture}$$\n\nWe take a smooth common resolution $Z$ of the birational maps $\\theta_i$. This gives birational morphisms $\\pi_i\\colon Z\\to X$ satisfying \\[CommonResZcommut\\]. ", "For each $i=1,\\ldots,t$, we apply Lemma \\[lem:ample models\\] to the morphism $Z/X_i$. This gives ample $\\Q$-divisors $\\Delta_i$ and $\\Delta_i'$ on $Z$ such that $\\pi_i$ is the ample model of $K_Z+\\Delta_i$, and $\\eta_i \\circ \\pi_i$ is the ample model of $K_Z+\\Delta_i'$. We then choose some large rational number $\\xi_i>0$ such that $K_Z+\\xi_i \\Delta_i$ and $K_Z+\\xi_i \\Delta_{i}'$ are ample, and then define $A_i=K_Z+\\xi_i \\Delta_i$ and $A_{t+i}=K_Z+\\xi_i\\Delta_i'$.\n\nThis provides ample $\\Q$-divisors $A_1,\\ldots,A_{2t}$ such that $K_Z+\\Delta_i$ and $K_Z+\\Delta_i'$ lie in $$\\Bigl\\{a_0 K_Z + \\sum_{i=1}^{2t} a_i A_i \\Bigm| a_0,\\ldots,a_{2t}\\ge 0 \\Bigr\\}\\cap \\widebar{\\Eff}(Z)$$ for each $i\\in \\{1,\\ldots,2t\\}$, proving \\[CommonResZisthere\\]. ", "Adding some additional ample divisors $A_j$, we can assume that $A_1,\\ldots,A_m$ generate the $\\R$-vector space $N^1(Z)$, giving \\[CommonResZgen\\].", "\n\nIn the following discussion (and until Corollary \\[cor:face\\_codim\\_3\\]) we work with the setting given by Proposition \\[pro:ResolutionZ\\], that is, the commutative diagram of Figure \\[fig:CommonZ\\] and an associated choice of cone $\\Cl \\subset N^1(Z)$. Also recall that $\\partial^+ \\Cl \\subset \\Cl$ is the subset of non-big divisors.", "\n\n\\[lem:disc\\_or\\_sphere\\] $\\partial^+\\Cl$ is the cone over a polyhedral complex homeomorphic to a disc or a sphere of dimension $\\rho(Z)-2$.\n\nConsider the auxiliary cone $\\Cl'$ of classes of the form $$\\sum a_i A_i \\text{ where } a_i \\ge 0 \\text{ for all } i.$$ In other words, $\\Cl'$ is the cone over the convex hull of the $A_i$, and in particular $\\Cl'$ is a closed subcone of the ample cone of $Z$. Let $\\partial^+ \\Cl'$ be the points in the boundary of $\\Cl'$ that are visible from the point $K_Z$. Formally: $$\\partial^+ \\Cl' = \\left\\lbrace D \\in \\Cl' \\mid [D, K_Z] \\cap \\Cl' = \\{D\\} \\right\\rbrace.$$ By an elementary convexity argument, this cone $\\partial^+\\Cl'$ is homeomorphic to the cone over a sphere or a disc of dimension $\\rho(Z)-2$, the first case occurring precisely if $-K_Z$ is in the interior of $\\Cl'$. Then we have a continuous map $$\\begin{tikzcd}[map]\n\\pi \\colon \\partial^+\\Cl' &\\to& \\partial^+\\Cl \\\\\nD &\\mapsto& \\pi(D)\n\\end{tikzcd}$$ that sends $D$ to the intersection of the segment $[D,K_Z]$ with $\\partial^+\\Cl$. The intersection exists because $K_Z \\not\\in \\widebar{\\Eff}(Z)$, while $D \\in \\Cl$, and the intersection is unique by convexity of $\\Cl$. The injectivity of $\\pi$ follows directly from the definition of $\\partial^+\\Cl'$, and $\\pi$ is also surjective, because by definition the cone $\\Cl$ in contained in the cone over the convex hull of $K_Z$ and the $A_i$, which is the same as the cone over the convex hull of $K_Z$ and $\\Cl'$. In conclusion $\\pi$ is a homeomorphism, as expected.", "\n\nRecall that the codimension of a face is taken relatively to the ambient space $N^1(Z)$, so in particular if $\\Fl^k \\subseteq \\partial^+\\Cl$ we have $k \\ge 1$.\n\nBy Remark \\[rem:codim1\\], a face $\\Fl^1$ of codimension 1 in $\\partial^+\\Cl$ corresponds to a Mori fibre space, or equivalently a rank 1 fibration (Lemma \\[lem:rank1=Mfs\\]). ", "More generally, we now prove that inner codimension $r$ faces in $\\partial^+\\Cl$ correspond to rank $r$ fibrations.", "\n\n\\[pro:boundary\\] Let $\\Fl^r \\subseteq \\partial^+ \\Cl$ be an inner codimension $r$ face. ", "By Lemma $\\ref{lem:faces_factory}$, we can write $\\Fl^r = \\widebar{\\Al_j}\\cap\\widebar{\\Al_i}$ with $\\Al_j$ a chamber of maximal dimension and $\\Al_i \\subseteq \\partial^+ \\Cl$ the Mori chamber containing the interior of $\\Fl^r$. Then\n\n1. ", " \\[boundary:rank\\] The associated morphism $\\phi_{ji}\\colon Z_j \\to Z_i$ is a rank $r$ fibration.", "\n\n2. ", " \\[boundary:through\\] If $\\Fl^s \\subseteq \\partial^+ \\Cl$ is an inner codimension $s$ face and $\\Fl^r\\subseteq\\Fl^s$, then the rank $r$ fibration associated to $\\Fl^r$ from \\[boundary:rank\\] factorises through the rank $s$ fibration associated to $\\Fl^s$.\n\n\\[boundary:rank\\] We check the assertions of Definition \\[def:rankFibration\\]:\n\n\\[fib:eta\\]. ", "By Lemma \\[lem:faces\\_factory\\]\\[faces:EinFji\\], $\\phi_{ji}\\colon Z_j \\to Z_i$ is a morphism with relative Picard rank equal to $r$, and $\\dim Z_i < \\dim Z_j$ because $\\Al_i\\subseteq \\partial^+ \\Cl$.\n\n\\[fib:singB\\]. ", "This is Proposition \\[pro:coneC\\]\\[coneC:singularities\\].", "\n\n\\[fib:big\\]. ", "To show that $-K_{Z_j}$ is $\\phi_{ji}$-big, we take $D\\in \\widebar{\\Al_j}\\cap\\Al_i$. By Proposition \\[pro:coneC\\], we have $D=K_Z+\\Delta$ for some ample divisor $\\Delta$, and $\\phi_{j*}D \\in \\Nef(Z_j)$ is $\\phi_{ji}$-trivial. ", "By Lemma \\[lem:semiampleample\\]\\[birsemiampleample\\] $\\phi_j$ is a semiample model of any element of $\\Al_j$. So $\\phi_j$ is a birational contraction and $\\phi_{j*} K_Z=K_{Z_j}$, which we rewrite as $-K_{Z_j}=\\phi_{j*}\\Delta-\\phi_{j*}D$. Since $\\Delta$ is ample and $\\phi_j$ is birational, the divisor $\\phi_{j*} \\Delta$ is big , which means we can write it as a sum of an ample and an effective divisor. ", "So $-K_{Z_j}$ is the sum of a $\\phi_{ji}$-ample and an effective divisor and hence is $\\phi_{ji}$-big by Lemma \\[lem:piBig\\].", "\n\n\\[fib:dream\\]. ", "We prove that $Z_j/Z_i$ is a Mori dream space:\n\n\\[dream:Q\\_factorial\\] and \\[dream:rat\\_connected\\]. ", "By Proposition \\[pro:coneC\\], $Z_j$ is $\\Q$-factorial terminal, $Z_i$ has rational singularities and $\\dim Z_j > \\dim Z_i$. A general fibre of $\\phi_{ji}$ has rational singularities by Lemma \\[lem:fibresAreTerminal\\]. ", "By Remark \\[rem:klt\\] we can assume that $(Z,\\Delta)$, and also $(Z_j,\\phi_{j*}\\Delta)$, are klt pairs. ", "By Proposition \\[pro:coneC\\]\\[coneC:phi\\_ji\\] the divisor $K_{Z_j}+\\phi_{j*} \\Delta=\\phi_{j*} D$ is $\\phi_{ji}$-trivial. ", "We have just seen that $-K_{Z_j}$ is $\\phi_{ji}$-big. ", "Then it follows from Lemma \\[lem:fibresRatConnected\\]\\[connected:-KBigNef\\] that a general fibre of $\\phi_{ji}$ is rationally connected.", "\n\n\\[dream:nef\\]. ", "We show that the nef cone $\\Nef(Z_j/Z_i)$ is generated by finitely many semiample divisors.", "\n\nWe take $D_j\\in \\Nef(Z_j)$ and set $D = \\phi_j^* D_j \\in N^1(Z)$. We choose $D' \\in \\interior{\\Fl}^r\\subseteq\\Al_i\\cap \\widebar{\\Al_j}$. By Remark \\[rem:InnerFace\\], for $t \\gg 0$ we have $D + t D' \\in \\Cl$. By Lemma \\[lem:complements\\]\\[comp:combination\\]&\\[comp:model\\], we have $D + tD' \\in\\widebar{\\Al_j}$. Since $\\phi_{j*}D'$ is $\\phi_{ji}$-trivial by Proposition \\[pro:coneC\\]\\[coneC:phi\\_ji\\]\\[AljAli\\], we get that $\\phi_{j*} (D+ t D') = D_j + t\\phi_{j*} D'$ is equivalent to $D_j$ in $\\Nef(Z_j/Z_i)$. Hence, any class in $\\Nef(Z_j/Z_i)$ can be represented by a divisor in $\\phi_{j*} \\widebar{\\Al_j}$. We conclude that $\\Nef(Z_j/Z_i)$ is generated by finitely many divisors of the form $\\phi_{j*}(K_Z+\\Delta)$, where $K_Z + \\Delta$ runs over the vertices of a polytope generating the cone $\\widebar{\\Al_j}$, and the $\\phi_{j*}(K_Z+\\Delta)$ are semiample by Proposition \\[pro:coneC\\]\\[coneC:phi\\_iBirational\\].", "\n\n\\[dream:mov\\]. ", "Let $D_j \\in \\IntMov(Z_j)$, in particular $D_j$ is big. ", "Set $D = \\phi_j^* D_j$ and pick $D'\\in\\interior{\\Fl}^r\\subseteq\\Al_i\\cap \\widebar{\\Al_j}$. By Remark \\[rem:InnerFace\\], for $t \\gg 0$ we have $\\hat D := D + t D' \\in \\Cl$.\n\nReplacing $D$ by an arbitrary close class in $\\Cl$ we can assume that $\\hat D = D + tD' \\in \\Al_k$ where $\\Al_k$ is of maximal dimension. ", "We also replace $D_j$ by $\\phi_{j*}D$, which is a small perturbation of the initial class hence still in $\\IntMov(Z_j)$. We keep the same notation for simplicity. (", "Observe that after perturbation we lose the property $D = \\phi_j^* D_j$, but we will not need it). ", "By finiteness of the chamber decomposition, $\\Al_k$ does not depend on the choice of the large real $t$, which also implies $D'\\in \\widebar{\\Al_k}\\cap\\Al_i$. So we have $\\Fl_{ji} \\subseteq \\Fl_{ki}$, hence a similar inclusion for the vector subspaces spanned by these faces. ", "By Lemma \\[lem:faces\\_factory\\]\\[faces:EinFji\\] this implies that all divisors contracted by $\\phi_j$ are also contracted by $\\phi_k$, hence $f_k:= \\phi_k \\circ \\phi_j^{-1}\\colon Z_j \\rat Z_k$ is a birational contraction.", "\n\nAs above $D_j$ and $\\hat D_j := D_j + t\\phi_{j*} D'$ represent the same class in $N^1(Z_j/Z_i)$. Moreover by Lemma \\[lem:semiampleample\\]\\[SemiampleFactorisation\\] we have $\\phi_{j*} D' = \\phi_{ji}^* D_i$, and the pull-back of an ample divisor being movable we have $\\phi_{j*} D' \\in \\widebar{\\Mov}(Z_j)$. So we have $\\hat D_j \\in \\IntMov(Z_j)$, and $\\phi_{j*} \\hat D = \\hat D_j$ with $\\hat D \\in \\Al_k$.\n\nWe now check that the birational contraction $f_k\\colon Z_j \\rat Z_k$ is the ample model of $\\hat D_j$. Since $\\hat D \\in \\Al_k$, we already have $\\hat D_k := f_{k*} \\hat D_j = \\phi_{k*} \\hat D \\in \\Ample(Z_k)$. Let $p\\colon W\\to Z$, $q_j\\colon W\\to Z_j$ and $q_k\\colon W\\to Z_k$ be common resolutions of $\\phi_j$ and $\\phi_k$ and $$\\begin{aligned}\nE:=p^* \\hat D-q_j^* \\hat D_j,&&\nF:=p^*\\hat D-q_k^*\\hat D_k\\end{aligned}$$ the $q_j$-exceptional and $q_k$-exceptional divisors respectively. ", "By Proposition \\[pro:coneC\\]\\[coneC:phi\\_iBirational\\] we have $F \\ge 0$. (On the other hand $E$ might not be effective). ", "We write $F = F_j + R$ where $F_j \\ge 0$ is $q_j$-exceptional and $R \\ge 0$ is $q_j$-effective. ", "Then we have $$\\begin{aligned}\nE - F_j &= q_k^* \\hat D_k - q_j^* \\hat D_j + R\\\\\n&= \\text{nef}- q_j\\text{-trivial} + q_j\\text{-effective} \\\\\n&= q_j\\text{-nef} + q_j\\text{-effective}.\\end{aligned}$$ The Negativity Lemma \\[lem:Negativity\\] gives $F_j - E \\ge 0$, hence also $q_j^* \\hat D_j-q_k^* \\hat D_k= F - E = R + F_j - E\\ge 0$, which by Lemma \\[lem:semiampleample\\]\\[birsemiampleample\\] achieves the proof that $f_k$ is the ample model of $\\hat D_j$.\n\nSince $\\hat D_j \\in \\IntMov(Z_j)$, its ample model $f_k$ is a pseudo-isomorphism. ", "Finally $\\hat D_j \\in f_k^*(\\Ample(Z_k/Z_i))$ where $Z_k$ is $\\Q$-factorial, and by taking closures we obtain $$\\widebar{\\Mov}(Z_j/Z_i) \\subseteq \\bigcup f_l^*(\\Nef(Z_l/Z_i))$$ for some finite collection of pseudo-isomorphisms $f_l\\colon Z_j \\ps Z_l$ over $Z_i$ to $\\Q$-factorial varieties.", "\n\nFor the other inclusion, we note that for any pseudo-isomorphism $f_l\\colon Z_j\\ps Z_l$ over $Z_i$, we have $f_l^*\\Ample(Z_l/Z_i)\\subset \\Mov(Z_l/Z_i)$ and the claim follows by taking closures.", "\n\n\\[fib:singX\\]. ", "Let $D_j\\in N^1(Z_j)$ be a divisor. ", "We now show that the output of any $D_j$-MMP from $Z_j$ over $Z_i$ can be obtained by running a $K_Z$-MMP from $Z$. Let $D'\\in\\Fl^r\\subseteq\\widebar{\\Al_j}$. Then by Proposition \\[pro:coneC\\]\\[coneC:phi\\_iBirational\\], $\\phi_j$ is a semiample model of $D'$, $\\phi_i$ is its ample model, and by Lemma \\[lem:semiampleample\\]\\[SemiampleFactorisation\\] $\\phi_{j*} D'=\\phi_{ji}^* H_i$ for some ample divisor $H_i$ on $Z_i$. To run a $D_j$-MMP from $Z_j$ over $Z_i$, we pick $H_j\\in\\Ample(Z_j)$ and consider all pseudo-effective convex combinations $D_t:=\\varepsilon(tD_j+(1-t)H_j)+\\phi_{ij}^* H_i$ for some $1\\gg\\varepsilon>0$. The set of the $\\phi_j^* D_t$ is a segment in a small neighborhood of $D'$ inside $\\Cl$. Therefore, any intermediate variety in this $D_j$-MMP over $Z_i$ can be obtained by running a $K_Z$-MMP from $Z$. In particular the output of this MMP has the form $\\Proj H^0(Z_j, D_{t_0}) = \\Proj H^0(Z, \\phi_j^* D_{t_0})$ for some $t_0 \\in (0,1)$, and by Proposition \\[pro:coneC\\]\\[coneC:MaxDim\\], this is a $\\Q$-factorial and terminal variety, as expected.", "\n\n\\[boundary:through\\] (Analogous to [@LZ Proposition 3.10(2)]): Let $\\Al_i,\\Al_k\\subseteq\\partial^+\\Cl$ be the chambers containing the interior of $\\Fl^r$, $\\Fl^s$ respectively. ", "By Lemma \\[lem:faces\\_factory\\]\\[faces:inner\\_is\\_inter\\] there exist maximal chambers $\\Al_j$ and $\\Al_l$ such that $\\Fl^r=\\widebar{\\Al_j}\\cap\\widebar{\\Al_i}$ and $\\Fl^s=\\widebar{\\Al_l}\\cap\\widebar{\\Al_k}$. Since moreover $\\Fl^r \\subseteq \\Fl^s$ implies that $\\widebar{\\Al_l} \\cap \\Al_i \\neq \\emptyset$, by Proposition \\[pro:coneC\\]\\[coneC:phi\\_ji\\] we have a commutative diagram induced by the maps from $Z$: $$\\begin{tikzcd}[link]\nZ_j \\ar[r,dashed] \\ar[dd] & Z_l \\ar[dd] \\ar[ddl] \\\\ \\\\\nZ_i & Z_k \\ar[l]\n\\end{tikzcd}$$ We want to prove that the birational map $Z_j \\rat Z_l$ is a birational contraction.", "\n\nLet $D\\in\\interior{\\Fl}^r \\subseteq \\Al_i$. There exists an ample class $\\Delta\\in\\Cl$ and $t_1 > 0$ such that $D = (1-t_1)\\Delta+t_1 K_Z$. For $t_1 > t_0 > 0$ sufficiently close to $t_0$, any chamber of maximal dimension $\\Al_{j_0}$ such that $ (1-t_0)\\Delta+t_0 K_Z \\in \\Al_{j_0}$ satisfies $\\Fl^r \\subset \\widebar{\\Al_{j_0}}$. Now there exists a small perturbation $\\Delta'$ of $\\Delta$ such that the segment $[\\Delta',K_Z]$ meets successively a chamber $\\Al_{j_0}$ and then the chamber $\\Al_l$. Indeed, $t_1>t_0$ and the ordering is preserved under a small perturbation. ", "Up to replacing $j$ by this $j_0$, by Remark \\[rem:MMPscaling\\] this segment corresponds to a $K_Z$-MMP with scaling of $\\Delta'$, and provides the expected birational contraction from $Z_j$ to $Z_l$.\n\n$$\\begin{aligned}\n\\begin{tikzpicture}[scale=4.5,font=\\footnotesize,baseline=(-K)]\n\\coordinate (E1) at (0:0) {};\n\\coordinate (E2) at (0:1) {};\n\\coordinate (P) at (60:1) {};\n\\coordinate (H) at (barycentric cs:E1=1,E2=1,P=1) {};\n\\coordinate (P1) at (barycentric cs:E1=0,E2=1,P=1) {};\n\\coordinate (P2) at (barycentric cs:E1=1,E2=0,P=1) {};\n\\coordinate (-K) at (barycentric cs:E1=2,E2=2,P=3) {};\n\\draw (E1) to [\"\\scriptsize $\\Fl^1_{15}$\",xshift=1] (P2)\n (P2) to [\"\\scriptsize $\\Fl^1_{35}$\",xshift=1] (P)\n (P) to [\"\\scriptsize $\\Fl^1_{36}$\",xshift=-1] (P1)\n (P1) to [\"\\scriptsize $\\Fl^1_{26}$\",xshift=-1] (E2)\n (E2) to [\"\\scriptsize $\\Fl^1_{47}$\",yshift =0.3mm] (E1);\n\\draw (E1) to [swap,\"\\scriptsize $\\Fl^1_{14}$\",yshift =1mm,xshift=-0.5](H)\n (E2) to [\"\\scriptsize $\\Fl^1_{24}$\",yshift =1mm,xshift=0.5](H)\n (H) to [\"\\scriptsize $\\Fl^1_{01}$\",yshift =1mm,xshift=0.5] (P2)\n (P1) to [\"\\scriptsize $\\Fl^1_{02}$\",yshift =1mm,xshift=-0.5] (H)\n (P1) to [\"\\scriptsize $\\Fl^1_{03}$\",swap,yshift=-.5] (P2);\n\\node at (H) {$\\bullet$};\n\\node at (P1) {$\\bullet$};\n\\node at (P2) {$\\bullet$};\n\\node at (E1) {$\\bullet$};\n\\node at (E2) {$\\bullet$};\n\\node at (P) {$\\bullet$};\n\\node[left] at (E1) {\\scriptsize $\\Fl^2_{17}$};\n\\node[left] at (P2) {\\scriptsize $\\Fl^2_{05}$};\n\\node[above] at (P) {\\scriptsize $\\Fl^2_{37}$};\n\\node[right] at (P1) {\\scriptsize $\\Fl^2_{06}$};\n\\node[right] at (E2) {\\scriptsize $\\Fl^2_{27}$};\n\\node[below,yshift=1] at (H) {\\scriptsize $\\Fl^2_{04}$};\n\\node[below,yshift=-18] at (H) {\\scriptsize $\\Fl^0_4\\!=\\!\\widebar{\\Al_4}$};\n\\node[below left,xshift=-10,yshift=3] at (H) {\\scriptsize $\\Fl^0_1\\!=\\!\\widebar{\\Al_1}$};\n\\node[above,yshift=20] at (-K) {\\scriptsize $\\Fl^0_3\\!=\\!\\widebar{\\Al_3}$};\n\\node[below right,xshift=10,yshift=3] at (H) {\\scriptsize $\\Fl^0_2\\!=\\!\\widebar{\\Al_2}$};\n\\node[above,yshift=5] at (H) {\\scriptsize $\\Fl^0_0\\!=\\!\\widebar{\\Al_0}$};\n\\end{tikzpicture}\n&&\n\\begin{tikzpicture}[scale=4.5,font=\\footnotesize,baseline=(-K)]\n\\coordinate (E1) at (0:0) {};\n\\coordinate (E2) at (0:1) {};\n\\coordinate (P) at (60:1) {};\n\\coordinate (H) at (barycentric cs:E1=1,E2=1,P=1) {};\n\\coordinate (P1) at (barycentric cs:E1=0,E2=1,P=1) {};\n\\coordinate (P2) at (barycentric cs:E1=1,E2=0,P=1) {};\n\\coordinate (-K) at (barycentric cs:E1=2,E2=2,P=3) {};\n\\draw (E1) to [\"\\scriptsize $Z_1/Z_5\\!=\\!\\p^1$\",xshift=1] (P2)\n (P2) to [\"\\scriptsize $Z_3/Z_5\\!=\\!\\p^1$\",xshift=1] (P)\n (P) to [\"\\scriptsize $Z_3/Z_6\\!=\\!\\p^1$\",xshift=-1] (P1)\n (P1) to [\"\\scriptsize $Z_2/Z_6\\!=\\!\\p^1$\",xshift=-1] (E2)\n (E2) to [\"\\scriptsize $Z_4/Z_7\\!=\\!\\pt$\",yshift =0.3mm] (E1);\n\\draw (E1) to (H)\n (E2) to (H)\n (H) to (P2)\n (P1) to (H)\n (P1) to (P2);\n\\node at (H) {$\\bullet$};\n\\node at (P1) {$\\bullet$};\n\\node at (P2) {$\\bullet$};\n\\node at (E1) {$\\bullet$};\n\\node at (E2) {$\\bullet$};\n\\node at (P) {$\\bullet$};\n\\node[below] at (E1) {\\scriptsize $Z_1/Z_7\\!=\\!\\pt$};\n\\node[left] at (P2) {\\scriptsize $Z_0/Z_5\\!=\\!\\p^1$};\n\\node[above] at (P) {\\scriptsize $Z_3/Z_7\\!=\\!\\pt$};\n\\node[right] at (P1) {\\scriptsize $Z_0/Z_6\\!=\\!\\p^1$};\n\\node[below] at (E2) {\\scriptsize $Z_2/Z_7\\!=\\!\\pt$};\n\\node[below,yshift=-18] at (H) {\\scriptsize $Z_4\\!=\\!\\p^2$};\n\\node[below left,xshift=-10,yshift=3] at (H) {\\scriptsize $Z_1\\!=\\!\\F_1$};\n\\node[above,yshift=20] at (-K) {\\scriptsize $Z_3\\!=\\!\\F_0$};\n\\node[below right,xshift=10,yshift=3] at (H) {\\scriptsize $Z_2\\!=\\!\\F_1$};\n\\node[above,yshift=5] at (H) {\\scriptsize $Z_0\\!=\\!Z$};\n\\end{tikzpicture}\\end{aligned}$$\n\n\\[ex:facesAndFib\\] On Figure \\[fig:facesAndFib\\] we label the boundary faces from Example \\[ex:P2blownup\\] with their corresponding rank $r$ fibration, as given by Proposition \\[pro:boundary\\] ($r = 1$ or $2$ here). ", "We also indicate the images of ample models corresponding to chambers of maximal dimension.", "\n\n\\[cor:windows\\_link\\] If the intersection $\\Fl_i^1\\cap\\Fl_j^1$ of non-big codimension $1$ faces has codimension $2$, then there is a Sarkisov link between the corresponding Mori fibre spaces.", "\n\nBy Proposition \\[pro:boundary\\] there is a rank $2$ fibration corresponding to the codimension $2$ face $\\Fl^2:=\\Fl_i^1\\cap\\Fl_j^1$ that factorises through the rank $1$ fibrations associated to $\\Fl_i^1$ and $\\Fl_j^1$. This is exactly the definition of a Sarkisov link (Definition \\[def:sarkisovLink\\]).", "\n\n\\[cor:face\\_codim\\_3\\] Let $\\Fl^3$ be a face in $\\partial^+\\Cl$ of codimension $3$ and $T/B$ be the associated rank $3$ fibration, as given in Proposition $\\ref{pro:boundary}$. Then the elementary relation associated to $T/B$ corresponds to the finite collection of codimension $1$ faces $\\Fl_1^1,\\dots,\\Fl_s^1$ containing $\\Fl^3$, and ordered such that $\\Fl_j^1$ and $\\Fl_{j+1}^1$ share a codimension $2$ face for all $j$ $($where indexes are taken modulo $s)$.\n\nThis is just a rephrasing of Proposition \\[pro:from T3\\], using Proposition \\[pro:boundary\\] to associate a rank $1$ or $2$ fibration dominated by $T/B$ to each codimension $1$ or $2$ face containing $\\Fl^3$, and using Corollary \\[cor:windows\\_link\\] to associate a Sarkisov link to each pair of codimension 1 faces sharing a common codimension 2 face.", "\n\nLet $X/B$ a Mori fibre space. ", "We denote by $\\BirMori(X)$ the groupoid of birational maps between Mori fibre spaces birational to $X$. The group of birational selfmaps $\\Bir(X)$ is a subgroupoid of $\\BirMori(X)$. The motivation for introducing the notion of elementary relation is the following result. ", "The first part is a reformulation of [@HMcK Theorem 1.1]. ", "The second part is strongly inspired by [@Kaloghiros Theorem 1.3], observe however that our notion of elementary relation is more restrictive.", "\n\nIn the statement we use the formalism of presentations by generators and relations for groupoids. ", "This is very similar to the more familiar setting of groups: we have natural notions of a free groupoid, and of a normal subgroupoid generated by a set of elements. ", "We refer to [@Brown §8.2 and 8.3] for details.", "\n\n\\[thm:sarkisov\\] Let $X/B$ be a terminal Mori fibre space.", "\n\n1. ", " \\[sarkisov1\\] The groupoid $\\BirMori(X)$ is generated by Sarkisov links and automorphisms.", "\n\n2. ", " \\[sarkisov2\\] Any relation between Sarkisov links in $\\BirMori(X)$ is generated by elementary relations.", "\n\n\\[sarkisov1\\] is the main result of [@HMcK]. ", "The idea of the proof is to take $Z$ a resolution of a given birational map $\\phi\\colon X_1/B_1 \\rat X_2/B_2$, and to consider the cone $\\Cl$ with a choice of ample divisors as given by Proposition \\[pro:ResolutionZ\\] (applied with $t = 2$, $\\theta_1 = \\phi$, $\\theta_2 = \\phi^{-1}$). ", "Then one takes a general 2-dimensional affine slice of $\\Cl$ that passes through the codimension 1 faces associated to $X_1/B_1$ and $X_2/B_2$. The intersection of this slice with $\\partial^+ \\Cl$ is a polygonal path corresponding to successive pairwise neighbour codimension 1 faces, and by Corollary \\[cor:windows\\_link\\] this gives a factorisation of $\\phi$ into Sarkisov links.", "\n\n\\[sarkisov2\\]. ", "The proof is essentially the same as in [@LZ Proposition 3.15], we repeat the argument for the convenience of the reader.", "\n\nLet $$X_0/B_0 \\stackrel{\\chi_1}\\longrat X_1/B_1 \\stackrel{\\chi_2}\\longrat \\dots \\stackrel{\\chi_t}\\longrat X_t/B_t$$ be a relation between $t$ Sarkisov links, meaning that $\\chi_t \\circ \\cdots \\circ \\chi_1$ is the identity on $X_0 = X_t$. We take a smooth resolution $Z$ dominating all the $X_i/B_i$, and consider the cone $\\Cl \\subset N^1(Z)$ constructed from a choice of ample divisors as in Proposition \\[pro:ResolutionZ\\]. ", "We may assume $\\rho(Z) \\ge 4$ (otherwise we simply blow-up some points on $Z$), so that by Lemma \\[lem:disc\\_or\\_sphere\\] the non-big boundary $\\partial^+ \\Cl$ is a cone over a polyhedral complex $\\Sl$ homeomorphic to a disc or a sphere of dimension $\\rho(Z) - 2 \\ge 2$. In particular, the section $\\Sl$ is simply connected. ", "Now we construct a 2-dimensional simplicial complex $\\Bl$ embedded in $\\Sl$ as follows. ", "Vertices are the barycenters $p(\\Fl^k)$ of codimension $k$ faces $\\Fl^k$ for $k = 1, 2$ or $3$. We call $k$ the type of the vertex. ", "We put an edge between $p(\\Fl^j)$ and $p(\\Fl^k)$ if $\\Fl^j$ is a proper face of $\\Fl^k$, and a 2-simplex for each sequence $\\Fl^3 \\subset \\Fl^2 \\subset \\Fl^1$. The complex $\\Bl$ is homeomorphic to the barycentric subdivision of the $2$-skeleton of the dual cell complex of $\\Sl$. It follows that $\\Bl$ is simply connected (recall that the 2-skeleton of a simply connected complex is again simply connected, see e.g. [@Hatcher Corollary 4.12]). ", "Then we restrict to the subcomplex $\\Il \\subseteq \\Bl$ corresponding to inner faces, which are the ones that intersect the relative interior of $\\Sl$. The simplicial complex $\\Il$ is a deformation retract of the interior of $\\Bl$, so $\\Il$ again is simply connected. ", "By Proposition \\[pro:boundary\\] we can associate a rank $r$ fibration to each vertex of type $r$, and two vertices are connected by an edge if and only if the corresponding fibrations factorise through each other. ", "By Corollary \\[cor:face\\_codim\\_3\\], around each vertex of type 3 there is a unique disc whose boundary loop encodes an elementary relation. ", "The complex $\\Il$ is a deformation retract of $\\Bl$ (hence also simply connected) and its $2$-dimensional components are the union of these discs. ", "The initial relation corresponds to a loop in $\\Il$ that only passes through vertices of types 1 and 2. ", "We can realise the homotopy of this loop to the constant loop inside the simply connected complex $\\Il$ by using these elementary relations, and this translates as a factorisation of the initial relation as a product of conjugates of elementary relations.", "\n\nThe whole construction leading to the previous theorem can be made in a relative setting, that is, where all involved varieties admit a morphism to a fixed base variety $B$. In fact the paper [@BCHM] on which relies [@HMcK] is written with this level of generality. ", "In the particular case where the base $B$ has dimension $n-1$, we obtain the following statement, slightly more precise than Theorem \\[thm:sarkisov\\]\\[sarkisov1\\].", "\n\n$$\\begin{tikzcd}[column sep=1.2cm,row sep=0.16cm]\nX=X_0\\ar[rrrr,dashed,\"\\phi\", bend left=13] \\ar[dd,\"\\eta_X\",swap] \\ar[r,dashed,\"\\chi_1\"] & X_1\\ar[dd] \\ar[r,dashed,\"\\chi_2\"] & \\cdots \\ar[r,dashed,\"\\chi_{t-1}\"]& X_{t-1} \\ar[r,dashed,\"\\chi_t\"]\\ar[dd] & X_t=Y\\ar[dd,\"\\eta_Y\"] \\\\ \\\\\nB_0=B\\ar[drr,\"\\id\",swap] & B_1\\ar[dr]& \\cdots& B_{t-1} \\ar[dl] & B_t=B\\ar[dll,\"\\id\"] \\\\\n&& B\n\\end{tikzcd}$$\n\n\\[lem:SarkisovOverBase\\] Let $\\eta_X\\colon X\\to B$ and $\\eta_Y\\colon Y\\to B$ be two conic bundles over the same base. ", "Then any birational map $\\phi\\colon X\\rat Y$ over $B$ decomposes into a sequence of Sarkisov links of conic bundles over $B$. More precisely, we have a commutative diagram as in Figure $\\ref{fig:SarkisovOverBase}$, such that for each $i=1,\\dots,t$, $B_i/B$ is a birational morphism, $X_i/B_i$ is a conic bundle and $\\chi_i$ is a Sarkisov link.", "\n\nExamples of elementary relations {#sec:examples}\n--------------------------------\n\nIn this section we give examples of elementary relations, mostly in dimension $n = 3$.\n\n\\[ex:kaloghiros\\] Let $X$ be a Fano variety with $\\Q$-factorial terminal singularities and Picard rank $3$. Then $X/\\pt$ is a rank 3 fibration (Example \\[ex:rank\\_r\\]\\[ex:rank\\_r:2\\]), hence there is an associated elementary relation. ", "In the case where $X$ is smooth of dimension $3$, these relations were studied systematically by Kaloghiros, using a classification result by Mori-Mukai: see [@Kaloghiros Example 4.9 and Figures 3,4 & 5]. ", "With respect to the setting of §\\[sec:kaloghiros\\], in these examples we have $Z = X$, $N^1(Z) \\simeq \\R^3$ and $\\partial^+\\Cl$ is the cone over a complex homeomorphic to a circle, which encodes the elementary relation. ", "Observe that the simple 2-dimensional Example \\[ex:facesAndFib\\] also belongs to this family of examples.", "\n\n\\[ex:P3\\_2lines\\] Let $L \\cup L' \\subset \\p^3$ be two secant lines, and $P$ the plane containing them. ", "Let $X \\to \\p^3$ be the blow-up of $L$ with exceptional divisor $E$, let $\\ell \\subset E$ be the fibre intersecting the strict transform of $L'$, and let $T \\to X$ be the blow-up of $L'$, with exceptional divisor $E'$.\n\nFrom $T$ we can flop $\\ell$ to get a 3-fold $T'$, which is obtained by the same two blow-ups in the reverse order: first the blow-up $X'\\to \\p^3$ of $L'\\subset \\p^3$ and then the blow-up $T'\\to X'$ of (the strict transform of) $L$ on $X'$.\n\nFrom $T$ or $T'$ one can contract the strict transform of $P$ onto a smooth point, obtaining two 3-folds $Y$ and $Y'$ also related by the flop of $\\ell$.\n\nThe elementary relation associated to the rank $3$ fibration $T/\\pt$ (or equivalently to $T'/\\pt$), is depicted on Figure \\[fig:relation\\_P3\\_2lines\\]. ", "There are five links in the relation, where $\\chi_1$ has type , $\\chi_2$ and $\\chi_4$ have type , $\\chi_3$ has type , and $\\chi_5$ has type .", "\n\n$$\\begin{tikzpicture}[scale=1.5,font=\\small]\n\\node (pt) at (90:0) {$\\pt$};\n\\node (P1) at (-90:\\Rc) {$\\p^1$};\n\\node (P3) at (90:\\Rc) {$\\p^3$};\n\\node (T) at (180:\\Ra) {$T$};\n\\node (T') at (0:\\Ra) {$T'$};\n\\node (X) at (170:\\Rb) {$X$};\n\\node (X') at (10:\\Rb) {$X'$};\n\\node (Y) at (-135:\\Rb) {$Y$};\n\\node (Y') at (-45:\\Rb) {$Y'$};\n\n\n\\draw[->] (P1) to (pt);\n\\draw[->] (P3) to (pt);\n\\draw[->] (X) to (P1);\n\\draw[->] (X) to [\"\\tiny $E$\",swap] (P3);\n\\draw[->] (Y) to (P1);\n\\draw[->] (T) to [\"\\tiny $E'$\"] (X);\n\\draw[->] (T) to [\"\\tiny $P$\",swap] (Y);\n\\draw[->] (X') to (P1);\n\\draw[->] (X') to [\"\\tiny $E'$\"] (P3);\n\\draw[->] (Y') to (P1);\n\\draw[->] (T') to [\"\\tiny $E$\",swap] (X');\n\\draw[->] (T') to [\"\\tiny $P$\"] (Y');\n\\draw[-,dotted] (T) to[bend left = 80,\"\\tiny flop\"] (T');\n\n\\draw[->,dashed] (P3) to [\"$\\chi_1$\", bend right=15,swap] (X);\n\\draw[->,dashed] (X) to [\"$\\chi_2$\", bend right=25] (Y);\n\\draw[->,dotted] (Y) to[bend right=20] node[above]{$\\chi_3$} node[below]{\\tiny flop} (Y');\n\\draw[->,dashed] (Y') to [\"$\\chi_4$\", bend right=25] (X');\n\\draw[->,dashed] (X') to [\"$\\chi_5$\", bend right=15,swap] (P3);\n\\end{tikzpicture}$$\n\n\\[ex:X3X2X1=1\\] Consider the blow-up $\\F_1 \\to \\p^2$ of a point, with exceptional curve $\\Gamma \\subset \\F_1$. In $\\p^1 \\times \\F_1$, write $D = \\p^1 \\times \\Gamma$, and $C = \\{0\\} \\times \\Gamma$. Let $T$ be the blow-up of $C$, with exceptional divisor $E$. Then $T/\\p^2$ is a rank 3 fibration, and we now describe the associated elementary relation (see Figure \\[fig:X3X2X1=1\\]). ", "We let the reader verify the following assertions (since all varieties are toric, one can for instance use the associated fans).", "\n\nFirst the two-rays game $T/\\F_1$ gives a link of type II $$\\chi_1\\colon \\p^1 \\times \\F_1 \\rat \\p^1 \\boxtimes \\F_1 ,$$ where $\\p^1 \\boxtimes \\F_1$ denotes a Mori fibre space over $\\F_1$ that is a non-trivial but locally trivial $\\p^1$-bundle. ", "The link $\\chi_1$ involves the pair $D \\cup E$ of divisors of type for $T/\\F_1$.\n\nThe divisor $D$ on $T$ can be contracted in two ways to a curve $\\p^1$, that is, $T$ dominates a flop between $\\p^1 \\boxtimes \\F_1$ and another variety $X$. This variety $X$ admits a divisorial contraction to $\\p^1 \\times \\p^2$, with exceptional divisor the strict transform of $E$, which here is a divisor of type for $X/\\p^2$. This corresponds to a link of type $$\\chi_2\\colon \\p^1 \\boxtimes \\F_1 \\rat \\p^1 \\times \\p^2.$$\n\nFinally the two-rays game $\\p^1 \\times \\F_1/\\p^2$, which factorises via $\\F_1$ and $\\p^1\\times\\p^2$, gives a link of type $$\\chi_3\\colon \\p^1 \\times \\p^2 \\rat \\p^1 \\times \\F_1.$$\n\nIn conclusion we get an elementary relation $\\chi_3 \\circ \\chi_2 \\circ \\chi_1 = \\id$.\n\nIn contrast with Lemma \\[lem:gonalityOfTypeII\\], observe that $D$ and $E$ are divisors of type for $T/\\F_1$, but divisors of type for $T/\\p^2$.\n\n$$\\begin{tikzpicture}[scale=1.5,font=\\small]\n\\node (T) at (180:\\Ra) {$T$};\n\\node (F1xP1) at (135:\\Rb) {$\\P^1 \\times \\F_1$};\n\\node (F1xP1bis) at (-135:\\Rb) {$\\P^1 \\boxtimes \\F_1$};\n\\node (X) at (-45:\\Rb) {$X$};\n\\node (P2) at (0:0cm) {$\\P^2$};\n\\node (F1) at (180:\\Rc) {$\\F_1$};\n\\node (P2xP1) at (0:\\Rc) {$\\p^1\\times \\p^2$};\n\\draw[->] (T) to [auto,\"\\tiny $E_{\\textup{I}} / E_{\\textup{II}}$\"] (F1xP1);\n\\draw[->] (T) to [auto,swap,\"\\tiny $D_{\\textup{II}}$\"] (F1xP1bis);\n\\draw[->] (T) to [bend right = 70,\"\\tiny $D_{\\textup{I}}$\",auto,swap] (X);\n\\draw[->] (F1xP1)-- (F1);\n\\draw[->] (F1xP1) to[\"\\tiny $D_{\\textup{I}}$\"] (P2xP1);\n\\draw[->] (F1xP1bis)-- (F1);\n\\draw[->] (X) to [auto,swap,pos=.3,\"\\tiny $E_{\\textup{I}}$\"] (P2xP1);\n\\draw[->] (F1)-- (P2);\n\\draw[->] (P2xP1)-- (P2);\n\\draw[dashed,->] (F1xP1) to [bend right=38] node[auto,swap]{$\\chi_1$} (F1xP1bis);\n\\draw[dashed,->] (F1xP1bis) to [bend right=42] node[auto,swap]{$\\chi_2$} (P2xP1);\n\\draw[dashed,->] (P2xP1) to [bend right=42] node[auto,swap]{$\\chi_3$} (F1xP1);\n\\draw[dotted,-] (F1xP1bis) to [bend right=20,\"\\tiny\\text{flop}\",swap] (X);\n\\end{tikzpicture}$$\n\n\\[ex:with\\_flip\\] Consider $\\p^1 \\times \\p^2$, and let $\\Gamma \\subset \\p^2$ be a line, $D \\simeq \\p^1 \\times \\Gamma$ the pull-back of $\\Gamma$ in $\\p^1 \\times \\p^2$, $\\Gamma' = \\{t\\} \\times \\Gamma \\subset D$ a section and $p \\in D \\setminus \\Gamma'$ a point. ", "Let $T \\to \\p^1 \\times \\p^2$ be the blow-up of $\\Gamma'$ and $p$, with respective exceptional divisors $D'$ and $E$, and denote again $D$ the strict transform of $\\p^1 \\times \\Gamma$ in $T$. Then the induced morphism $\\eta\\colon T \\to \\p^2$ is a rank 3 fibration that gives rise to the relation on Figure \\[fig:flip\\].", "\n\nThe figure was computed using toric fans in $\\Z^3$, starting from the standard fan of $\\p^1 \\times \\p^2$ with primitive vectors $(1,0,0)$, $(0,1,0)$, $(-1,-1,0)$, $(0,0,1)$, $(0,0,-1)$, and with the following choices $$\\begin{aligned}\nD: (1,0,0), && D':(1,0,1), && E: (1,1,-1).\\end{aligned}$$ The varieties $T'$ and $W'$ both have one terminal singularity, all other varieties are smooth. ", "There are two distinct Francia flips from $T'$, which are $T' \\ps T$ and $T' \\ps T''$. Observe also that the link $\\chi_1$ is exactly Example \\[ex:simple\\_links\\]\\[simple\\_link:2\\].", "\n\n$$\\begin{tikzpicture}[scale=1.4,font=\\small]\n\\node (P2) at (0:0) {$\\P^2$};\n\\node (T) at (0:\\Ra) {$T$};\n\\node (T') at (120:\\Ra) {$T'$};\n\\node (T'') at (-120:\\Ra) {$T''$};\n\\node (X) at (55:\\Rb) {$X$};\n\\node (W') at (150:\\Rb) {$W'$};\n\\node (W) at (-150:\\Rb) {$W$};\n\\node (P1xF1) at (-90:\\Rb) {$\\p^1 \\boxtimes \\F_1$};\n\\node (Y) at (-55:\\Rb) {$Y$};\n\\node (F1) at (-120:\\Rc) {$\\F_1$};\n\\node (P1xP2) at (0:\\Rc) {$\\p^1\\times \\p^2$};\n\\node (P1xP2bis) at (120:\\Rc) {$\\p^1\\boxtimes \\p^2$};\n\\draw[->] (F1) to (P2);\n\\draw[->] (P1xP2) to (P2);\n\\draw[->] (P1xP2bis) to (P2);\n\\draw[->] (P1xF1) to (F1);\n\\draw[->] (W) to (F1);\n\\draw[->] (T) to [\"$E$\",swap] (X);\n\\draw[->] (T') to [\"$D$\",swap] (W');\n\\draw[->] (T) to [\"$D'$\"] (Y);\n\\draw[->] (T'') to [\"$D$\"] (W);\n\\draw[->] (T'') to [\"$D'$\",swap] (P1xF1);\n\\draw[->] (X) to [\"$D'$\"] (P1xP2);\n\\draw[->] (X) to [\"$D$\",swap] (P1xP2bis);\n\\draw[->] (W') to [\"$E$\",swap] (P1xP2bis);\n\\draw[->] (Y) to [\"$E$\",swap] (P1xP2);\n\\draw[->,dotted] (T') to [\"\\tiny flip\",bend left=50] (T);\n\\draw[->,dotted] (T') to [\"\\tiny flip\",swap, bend right=50] (T'');\n\\draw[->,dotted] (W') to [\"\\tiny flip\",swap] (W);\n\\draw[dotted] (Y) to [\"\\tiny flop\", pos=1] (P1xF1);\n\\draw[->,dashed] (P1xP2) to [\"$\\chi_1$\", bend right=20] (P1xP2bis);\n\\draw[->,dashed] (P1xP2bis) to [\"$\\chi_2$\", bend right=20] (W);\n\\draw[->,dashed] (W) to [\"$\\chi_3$\", bend right=20] (P1xF1);\n\\draw[->,dashed] (P1xF1) to [\"$\\chi_4$\", bend right=20] (P1xP2);\n\\end{tikzpicture}$$\n\n\\[ex:AZ\\] The article [@AZ2017] contains a beautiful example of an elementary relation involving five Sarkisov links. ", "In Figure \\[fig:relation\\_AZ\\] we reproduce the diagram from [@AZ2017 §5.2], and we refer to their paper for a detailed description of the varieties. ", "The Sarkisov links $\\chi_1$ and $\\chi_3$ have type , $\\chi_2$ has type , $\\chi_4$ has type and $\\chi_5$ type . ", "The relation is associated to the rank 3 fibration $Z_1'/\\pt$, or equivalently to $Z_2'/\\pt$. In fact other equivalent choices of varieties of Picard rank 3 are omitted from the picture (dominating respectively $Y_1'$, $X_3'$, $X_1'$ and $X_1''$). ", "The morphisms from $Z, \\bar Z$ and $\\tilde Z$ to $\\P^1$ are fibrations in cubic surfaces. ", "Observe that the top rows of the Sarkisov diagrams display non trivial pseudo-isomorphisms, involving flips and flops. ", "Note that each pseudo-isomorphism labeled “$n$ flops” really corresponds to a single flop with $n$ components (which by definition are all numerically proportional), in accordance with Remark \\[rem:top\\_row\\].", "\n\n$$\\begin{tikzpicture}[scale=1.5,font=\\small]\n\\node (pt) at (0:0) {$\\pt$};\n\\node (P1) at (0:\\Rc) {$\\p^1$};\n\\node (Y) at (120:\\Rc) {$Y$};\n\\node (X) at (-120:\\Rc) {$X$};\n\\node (barZ) at (-15:\\Rb) {$\\bar Z$};\n\\node (Z) at (15:\\Rb) {$Z$};\n\\node (Y1') at (105:\\Rb) {$Y_1'$};\n\\node (X3') at (135:\\Rb) {$X_3'$};\n\\node (X1') at (-135:\\Rb) {$X_1'$};\n\\node (X1'') at (-105:\\Rb) {$X_1''$};\n\\node (X2') at (180:\\Rb) {$X_2'$};\n\\node (tildeZ) at (-45:\\Rb) {$\\tilde Z$};\n\\node (X2'') at (-75:\\Rb) {$X_2''$};\n\\node (Z1') at (-45:\\Ra) {$Z_1'$};\n\\node (Z2') at (-15:\\Ra) {$Z_2'$};\n\\draw[->] (P1) to (pt);\n\\draw[->] (Y) to (pt);\n\\draw[->] (X) to (pt);\n\\draw[->] (Z) to (P1);\n\\draw[->] (barZ) to (P1);\n\\draw[->] (tildeZ) to (P1);\n\\draw[->] (Y1') to (Y);\n\\draw[->] (X3') to (Y);\n\\draw[->] (X1') to (X);\n\\draw[->] (X1'') to (X);\n\\draw[->] (Z1') to (tildeZ);\n\\draw[->] (Z2') to (barZ);\n\\draw[dotted,-] (X1') to [bend left=15,\"\\tiny $11$ flops\"] (X2');\n\\draw[dotted,->] (X2') to [bend left=15,\"\\tiny flip\"] (X3');\n\\draw[dotted,-] (Y1') to [bend left=30,\"\\tiny $9$ flops\"] (Z);\n\\draw[dotted,-] (X1'') to [bend right=12,\"\\tiny $7$ flops\",swap] (X2'');\n\\draw[dotted,->] (X2'') to [bend right=12,\"\\tiny flip\",swap] (tildeZ);\n\\draw[dotted,-] (Z1') to [bend right=12,\"\\tiny $6$ flops\",swap] (Z2');\n\\draw[->,dashed] (Y) to [\"$\\chi_1$\", bend right=35,swap] (X);\n\\draw[->,dashed] (X) to [\"$\\chi_2$\", bend right=25] (tildeZ);\n\\draw[->,dashed] (tildeZ) to[\"$\\chi_3$\", bend right=15] (barZ);\n\\draw[->] (barZ) to[bend right=12] node[left]{\\tiny $\\simeq$} node[right]{$\\chi_4$} (Z);\n\\draw[->,dashed] (Z) to [\"$\\chi_5$\", bend right=25,swap] (Y);\n\\end{tikzpicture}$$\n\nElementary relations involving Sarkisov links of conic bundles of type {#sec:constraints}\n=======================================================================\n\nThis section is devoted to the study of elementary relations involving Sarkisov links of conic bundles of type that are complicated enough, meaning their covering gonality is large. ", "We give some restriction on such relations that will allow us to prove Theorem \\[TheoremBirMori\\]. ", "Firstly in Proposition \\[pro:boundOnGenus\\] we cover the case of relations over a base of dimension $\\le n- 2$, where $n$ is the dimension of the Mori fibre spaces, using the BAB conjecture and working with Sarkisov links of large enough covering gonality. ", "Secondly, the case of relations over a base of dimension $n-1$ is handled in Proposition \\[pro:X4X3X2X1=id\\], using only the assumption that the covering gonality is $>1$.\n\nA consequence of the BAB conjecture {#sec:BAB}\n-----------------------------------\n\nThe following is a consequence of the BAB conjecture, which was recently established in arbitrary dimension by C. Birkar.", "\n\n\\[pro:BAB\\] Let $n$ be an integer, and let $\\Ql$ be the set of weak Fano terminal varieties of dimension $n$. There are integers $d,l,m\\ge 1$, depending only on $n$, such that for each $X \\in \\Ql$ the following hold:\n\n1. ", " $\\dim(H^0(-mK_X))\\le l$;\n\n2. ", " The linear system $\\lvert -mK_X \\rvert$ is base-point free;\n\n3. ", " \\[BAB:3\\] The morphism $\\phi\\colon X\\stackrel{\\lvert -mK_X\\rvert}{\\longto} \\p^{\\dim(H^0(-mK_X))-1}$ is birational onto its image and contracts only curves $C\\subseteq X$ with $C\\cdot K_X=0$;\n\n4. ", " $\\deg \\phi(X) \\le d$.\n\nBy [@BirkarS Theorem 1.1], varieties in $\\Ql$ form a bounded family (here we use the observation that for a given $X \\in \\Ql$, the pair $(X,\\emptyset)$ is $\\eps$-lc for any $0 < \\eps < 1$). ", "In particular, by [@BirkarA Lemma 2.24], the Cartier index of such varieties is uniformly bounded. ", "Then [@Kollar93 Theorem 1.1] gives the existence of $m = m(n)$ such that $\\lvert -mK_X \\rvert$ is base-point free for each $X \\in \\Ql$. By [@BirkarA Theorem 1.2], we can increase $m$ if needed, and assume that the associated morphism $$\\begin{tikzcd}[column sep=1.5cm]\n\\phi\\colon X \\ar[r,\"\\lvert -mK_X\\rvert\"] & \\p^{\\dim(H^0(-mK_X))-1}\n\\end{tikzcd}$$ is birational onto its image. ", "As it is a morphism, this implies that it contracts only curves $C\\subseteq X$ with $C\\cdot K_X=0$. Finally, since $\\Ql$ is a bounded family, the two integers $\\dim(H^0(-mK_X))$ and $\\deg \\phi(X)$ are bounded.", "\n\n\\[cor:BAB\\] Let $\\pi\\colon Y \\to X$ be the blow-up of a reduced but not necessarily irreducible codimension $2$ subvariety $\\Gamma \\subset X$, $Y \\ps \\hat Y$ a pseudo-isomorphism, and assume that both $X$ and $\\hat Y$ are weak Fano terminal varieties of dimension $n \\ge 3$, whose loci covered by curves with trivial intersection against the canonical divisor has codimension at least $2$. Let $\\phi$ be the birational morphism associated to the linear system $\\lvert -mK_X \\rvert$, with $m$ given by Proposition $\\ref{pro:BAB}$, and assume that $\\Gamma$ is not contained in the exceptional locus $\\Ex(\\phi)$. Then through any point of $\\Gamma \\setminus \\Ex(\\phi)$ there is an irreducible curve $C\\subseteq \\Gamma$ with $\\gon(C)\\le d$, where $d$ is the integer from Proposition $\\ref{pro:BAB}$.\n\nWe choose the integers $d,l,m\\ge 1$ associated to the dimension $n$ in Proposition \\[pro:BAB\\]. ", "We write $a=\\dim(H^0(-mK_{X}))-1$ and $b=\\dim(H^0(-mK_{Y}))-1$. Using the pseudo-isomorphism $Y \\ps \\hat{Y}$, we also have $b=\\dim(H^0(-mK_{\\hat{Y}}))-1$. By Proposition \\[pro:BAB\\] the morphisms given by the linear systems $\\lvert -mK_{X}\\rvert$ and $\\lvert -mK_{\\hat{Y}}\\rvert$ are birational onto their images and are moreover pseudo-isomorphisms onto their images, because of the assumption that the locus covered by curves with non-positive intersection against the canonical divisor has codimension at least 2.", "\n\nSince $Y\\to X$ is the blow-up of $\\Gamma$, each effective divisor equivalent to $-mK_{Y}$ is the strict transform of an effective divisor equivalent to $-mK_{X}$ passing through $\\Gamma$ (with some multiplicity). ", "In particular, we have $b\\le a$ and obtain a commutative diagram $$\\begin{tikzcd}[column sep=1.6cm,row sep=0.3cm]\nX \\ar[d,\"\\lvert -mK_{X}\\rvert\"] \\ar[d,\"\\phi\",swap]\n& Y\\ar [rd,\"\\lvert -mK_{Y}\\rvert\",pos=.25,dashed,swap] \\ar[l]\\ar[rr,dotted,-]\n&&\\hat{Y} \\ar[ld,\"\\lvert -mK_{\\hat{Y}}\\rvert\",pos=.25]\\\\\n\\p^{a}\\ar[rr,\"\\pi\",dashed,swap]&&\\p^{b}\n\\end{tikzcd}$$ where $\\pi$ is a linear projection away from a linear subspace $\\mathcal{L}\\simeq \\p^r$ of $\\p^{a}$ containing the image of $\\Gamma$. Recall that we write $\\phi\\colon X\\to \\p^{a}$ the morphism given by $\\lvert -mK_{X}\\rvert$. The variety $\\phi(X) \\subseteq \\p^a$ has dimension $n$ and degree $\\le d$ (Proposition \\[pro:BAB\\]), and is not contained in a hyperplane section. ", "Since by assumption $\\Gamma \\subsetneq \\Ex(\\phi)$, we get that $\\phi$ induces a birational morphism from $\\Gamma$ to $\\phi(\\Gamma)$.\n\nWe now prove that there is no (irreducible) variety $S\\subseteq \\phi(X)\\cap \\mathcal{L}$ of dimension $n-1$ (recall that $\\phi(\\Gamma)\\subseteq \\phi(X)\\cap \\mathcal{L}$ has dimension $n-2$). ", "Indeed, otherwise the strict transform of $S$ on $X$ would be a variety $S_X\\subset X$ birational to $S$, so its strict transform in $\\hat Y$, and in $\\p^b$ is again birational to $S$ (as the birational map from $Y$ to its image in $\\p^b$ is a pseudo-isomorphism). ", "The linear system of the rational map $X\\rat \\p^b$ is obtained from the linear system associated to $X\\rat \\p^a$ by taking the subsystem associated to hyperplanes through $\\mathcal{L}$. Hence, if $S\\subseteq \\mathcal{L}$, then every element of the linear system $\\lvert-mK_{Y}\\rvert$ contains the strict transform $S_Y$ of $S$ in $\\hat Y$. This is impossible, as $\\lvert-mK_{\\hat Y}\\rvert$ is base-point free (Proposition \\[pro:BAB\\]).", "\n\nNow, the fact that $\\phi(X)\\cap \\mathcal{L}\\subseteq\\p^a$ does not contain any variety of dimension $\\ge n-1$ implies, by Bézout Theorem, that all its irreducible components of dimension $n-2$ have degree $\\le d$. Therefore, each of the irreducible components of $\\phi(\\Gamma)$ (birational to $\\Gamma$) has degree $\\le d$.\n\nWe are now able to finish the proof, by showing that through any point $q\\in\\Gamma \\setminus \\Ex(\\phi)$ there is an irreducible curve $C\\subseteq \\Gamma$ with $\\gon(C)\\le d$. Since $\\Gamma\\to \\phi(\\Gamma)$ is a local isomorphism at $q$, it suffices to take a general linear projection from $\\p^a$ to a linear subspace of dimension $n-2$, and to take $C$ equal to the preimage of a line through the image of $\\phi(q)$.\n\n\\[pro:boundOnGenus\\] For each dimension $n\\ge 3$, there exists an integer $d_n\\ge 1$ depending only on $n$ such that the following holds. ", "If $\\chi$ is a Sarkisov link of conic bundles of type that arises in an elementary relation induced by rank $3$ fibration $T/B$ with $\\dim(T)=n$ and $\\dim(B)\\le n-2$, then $\\covgon(\\chi)\\le \\max\\{d_n,8\\conngon(T)\\}$.\n\nWe choose $d_n\\ge 8$ to be bigger than the integers $d$ given by Proposition \\[pro:BAB\\] for the dimensions $3,\\dots,n$, and prove the result for this choice of $d_n$.\n\nThe Sarkisov link $\\chi$, which is dominated by $T/B$ by assumption, has the form $$\\begin{tikzcd}[link]\n Y_1\\ar[dd]\\ar[r,dotted,-]& T \\ar[r,dotted,-] & Y_2\\ar[dd] \\\\ \\\\\nX_1 \\ar[rr,\"\\chi\",dashed] \\ar[rd]&& X_2 \\ar[ld]\\\\\n&\\tilde{B}\\ar[dd]\\\\ \\\\\n&B\n\\end{tikzcd}$$ where $X_1,X_2,Y_1,Y_2$ have dimension $n$ and $\\tilde{B}$ has dimension $n-1$. Since $\\dim B \\le n-2$, we have $\\rho(\\tilde{B}/B) \\ge 1$, and on the other hand $\\rho(Y_i/B) \\le 3$, for $i=1,2$, which implies that $\\rho(\\tilde{B}/B) = 1$, and that the birational contractions $T\\ps Y_1$, $T\\ps Y_2$ are pseudo-isomorphisms. ", "Moreover, $Y_1\\to X_1$ contracts a divisor $E$ onto a variety $\\Gamma_1\\subset X_1$ of dimension $n-2$, birational to its image $\\tilde{\\Gamma}\\subset \\tilde{B}$ via the morphism $X_1\\to \\tilde{B}$ (Lemma \\[lem:SarkiIIConic\\]). ", "We need to check that $\\covgon(\\Gamma_1)=\\covgon(\\tilde{\\Gamma})\\le d_n$, where $d_n$ is chosen as above. ", "We may then assume that $\\covgon(\\tilde{\\Gamma})>1$.\n\nNow, $\\tilde B/B$ is a klt Mori fibre space by Lemma \\[lem:B’toB\\] and $X_1/B$ is a rank $2$ fibration by Lemma \\[lem:StableUnderMMP\\]\\[stab\\_under\\_MMP:1\\]. ", "By Lemma \\[lem:weakFano\\], the rank $2$ fibration $X_1/B$ is pseudo-isomorphic, via a sequence of log-flips over $B$, to another rank $2$ fibration $X/B$ such that $-K_{X}$ is relatively nef and big over $B$. We then use Lemma \\[lem:2raysOverFlip\\] to obtain a sequence of log-flips $Y_1\\ps Y$ over $B$ such that the induced map $Y\\to X$ is a divisorial contraction. ", "By Lemma \\[lem:weakFano\\] again, we get a sequence of log-flips over $B$ from $Y/B$ to another rank $3$ fibration $\\hat{Y}/B$ such that $-K_{\\hat{Y}}$ is relatively nef and big over $B$. $$\\begin{tikzcd}[link]\n Y_1\\ar[dd]\\ar[r,dotted,-]&Y\\ar[dd]\\ar[r,dotted,-]& \\hat{Y}\\ar[ldddd] \\\\ \\\\\nX_1 \\ar[rdd]\\ar[r,dotted,-]& X\\ar[dd]\\\\ \\\\\n&B\n\\end{tikzcd}$$ As $\\covgon(\\tilde{\\Gamma})>1$, by Lemma \\[lem:fibresRatConnected\\]\\[connected:flips\\] the codimension $2$ subvariety $\\Gamma_1\\subset X_1$ is not contained in the base-locus of the pseudo-isomorphism $X_1\\ps X$. So the image $\\Gamma\\subset X$ of $\\Gamma_1$ is birational to $\\Gamma_1$, and it suffices to show that $\\covgon(\\Gamma)\\le d_n$. Observe that $Y\\to X$ is the blow-up of $\\Gamma$ (Lemma \\[lem:divContToCodim2\\]).", "\n\nWe take a general point $p\\in B$, and consider the fibres over $p$ in $X$, $Y$ and $\\hat{Y}$ respectively, that we denote by $X_p$, $Y_p$ and $\\hat{Y}_p$, and which are varieties of dimension $$n_0=n-\\dim B \\in \\{ 2,\\dots,n\\}.$$\n\nBy Corollary \\[cor:weakFanoFiber\\] the two varieties $X_p$ and $\\hat{Y}_p$ are weak Fano terminal varieties. ", "Moreover, $Y_p$ and $\\hat{Y}_p$ are pseudo-isomorphic, as $Y\\ps \\hat{Y}$ is a sequence of log-flips over $B$.\n\nObserve that $\\tilde{\\Gamma}\\subset \\tilde{B}$ is a hypersurface and that $\\tilde{\\Gamma}\\to B$ is surjective. ", "Indeed, otherwise $\\tilde{\\Gamma}$ would be the preimage of a divisor on $B$, and we would have $\\covgon(\\tilde{\\Gamma}) = 1$, as the preimage of each point of $\\tilde{B}\\to B$ is covered by rational curves (Lemma \\[lem:B’toB\\]), in contradiction with our assumption. ", "This implies that the morphism $\\Gamma\\to B$ induced by the restriction of $X/B$ is again surjective.", "\n\nWe then denote by $\\Gamma_p\\subset X_p$ the codimension $2$ subscheme $\\Gamma_p=\\Gamma\\cap X_p$, which is the fibre of $\\Gamma\\to B$ over $p$, and which is not necessarily irreducible. ", "Observe that $Y_p\\to X_p$ is the blow-up of $\\Gamma_p$, as $Y\\to X$ is the blow-up of $\\Gamma$ (as explained before) and because the fibre over $p$ is transverse to $\\Gamma$ (Lemma \\[lem:SarkiIIConic\\]\\[SarkiII:3\\]).", "\n\nSuppose first that $n_0=2$, which corresponds to $\\dim(\\Gamma)=\\dim(B)$. In this case, $X_p$ and $Y_p\\simeq \\hat{Y}_p$ are smooth del Pezzo surfaces, because by Corollary \\[cor:weakFanoFiber\\] the locus covered by curves trivial against the canonical divisor has codimension 2, hence is empty in the case $n_0 = 2$. Moreover $\\Gamma_p$ is a disjoint union of $r$ points, where $r$ is the degree of the field extension $\\C(B)\\subseteq \\C(\\Gamma_1)$. As $Y_p$ is obtained from $X_p$ by blowing-up $\\Gamma_p$, the degree of the field extension is at most $8$, which implies that $\\covgon(\\Gamma)\\le 8\\cdot \\covgon(B)\\leq8\\conngon(T)$ (Lemma \\[lem:CovgonRatMorphism\\]).", "\n\nWe now consider the case $n_0\\ge 3$, which implies that $\\Gamma_p$ has dimension $n_0-2\\ge 1$. We apply Corollary \\[cor:BAB\\] to the blow-up $Y_p\\to X_p$ of $\\Gamma_p$ and the pseudo-isomorphism $Y_p\\ps \\hat{Y}_p$. The fact that the loci on $X_p$ or $\\hat Y_p$ covered by curves with trivial intersection against the canonical divisor has codimension at least 2 follows from Corollary \\[cor:weakFanoFiber\\]. ", "We obtain that for a general $p$, $\\Gamma_p \\setminus \\Ex(\\phi)$ is covered by curves of gonality at most $d_n$. In conclusion, we have found an open set $U = \\Gamma \\setminus \\Ex(\\phi) \\subseteq \\Gamma$ covered by curves of gonality at most $d_n$, as expected.", "\n\n\\[rem:BAB\\] It is not clear to us whether Proposition \\[pro:boundOnGenus\\] could also hold for a link $\\chi$ of type between arbitrary Mori fibre spaces.", "\n\nFor instance in the case of threefolds, if $\\chi$ is a link of type between del Pezzo fibrations that starts with the blow-up a curve of genus $g$ contained in one fibre, we suspect that $g$ cannot be arbitrary large but we are not aware of any bound in the literature.", "\n\nSome elementary relations of length 4\n-------------------------------------\n\n\\[pro:X4X3X2X1=id\\] Let $\\chi_1$ be a Sarkisov link of conic bundles of type with $\\covgon(\\chi_1) > 1$. Let $T/B$ be a rank $3$ fibration with $\\dim B = \\dim T -1$, which factorises through the Sarkisov link $\\chi_1$. Then, the elementary relation associated to $T/B$ has the form $$\\chi_4\\circ \\chi_3\\circ \\chi_2\\circ \\chi_1=\\id,$$ where $\\chi_3$ is a Sarkisov link of conic bundles of type that is equivalent to $\\chi_1$.\n\n$$\\begin{aligned}\n\\begin{tikzpicture}[scale=1.1,font=\\small,outer sep=-1pt]\n\\node (Y1) at (60+90:\\Ra) {$Y_1$};\n\\node (Y2) at (120+90:\\Ra) {$Y_2$};\n\\node (T3') at (180+90:\\Ra) {$T_3'$};\n\\node (T3) at (240+90:\\Ra) {$T_3$};\n\\node (T4) at (300+90:\\Ra) {$T_4$};\n\\node (T4') at (360+90:\\Ra) {$T_4'$};\n\\node (X1) at (60+90:\\Rb) {$X_1$};\n\\node (X2) at (120+90:\\Rb) {$X_2$};\n\\node (X3) at (300:\\Rc) {$X_3$};\n\\node (X4) at (420:\\Rc) {$X_4$};\n\\node (Y3') at (180+90:\\Rb) {$Y_3'$};\n\\node (Y3) at (240+90:\\Rb) {$Y_3$};\n\\node (Y4) at (300+90:\\Rb) {$Y_4$};\n\\node (Y4') at (360+90:\\Rb) {$Y_4'$};\n\\node (B) at (0:0cm) {$B$};\n\\node (Bhat) at (180:\\Rc) {$\\hat{B}$};\n\\draw[dotted,-] (Y1) to [bend right=21] (Y2)\n (Y2) to [bend right=21] (T3')\n (T3') to [bend right=21] (T3)\n (T3) to [bend right=21] (T4)\n (T4) to [bend right=21] (T4')\n (T4') to [bend right=21] (Y1)\n (X2) to [bend right=21] (Y3')\n (Y3) to [bend right=21] (Y4)\n (Y4') to [bend right=21] (X1);\n\\draw[->] (Bhat) to(B);\n\\draw[->] (X3) to(B);\n\\draw[->] (X4) to(B);\n\\draw[->] (T4') to [auto,\"\\tiny $E_1$\"] (Y4');\n\\draw[->] (Y1) to [auto,pos=.1,\"\\tiny $E_1$\"] (X1);\n\\draw[->] (Y2) to [auto,swap,pos=.1,\"\\tiny $E_2$\"] (X2);\n\\draw[->] (T3') to [auto,swap,\"\\tiny $E_2$\"] (Y3');\n\\draw[->] (T3) to [auto,swap,\"\\tiny $G$\"] (Y3);\n\\draw[->] (T4) to [auto,\"\\tiny $G$\"] (Y4);\n\\draw[->] (Y3') to [auto,swap,\"\\tiny $G$\"] (X3);\n\\draw[->] (Y4') to [auto,\"\\tiny $G$\"] (X4);\n\\draw[->] (Y3) to [auto,swap,pos=.7,\"\\tiny $E_2$\"] (X3);\n\\draw[->] (Y4) to [auto,pos=.7,\"\\tiny $E_1$\"] (X4);\n\\draw[->] (X1) to (Bhat);\n\\draw[->] (X2) to (Bhat);\n\\draw[dashed,->] (X1) to [bend right=30,auto,swap,\"$\\chi_1$\"] (X2);\n\\draw[dashed,->] (X2) to [auto,swap,pos=.3,\"$\\chi_2$\"] (X3);\n\\draw[dashed,->] (X3) to [bend right=38,auto,swap,\"$\\chi_3$\"] (X4);\n\\draw[dashed,->] (X4) to [auto,swap,pos=.7,\"$\\chi_4$\"] (X1);\n\\end{tikzpicture}\n&&\n\\begin{tikzpicture}[scale=1.1,font=\\small,outer sep=-1pt]\n\\node (Y1) at (135:\\Ra) {$Y_1$};\n\\node (Y2) at (-135:\\Ra) {$Y_2$};\n\\node (Y3) at (-45:\\Ra) {$Y_3$};\n\\node (Y4) at (45:\\Ra) {$Y_4$};\n\\node (X1) at (135:\\Rb) {$X_1$};\n\\node (X2) at (-135:\\Rb) {$X_2$};\n\\node (X3) at (-45:\\Rb) {$X_3$};\n\\node (X4) at (45:\\Rb) {$X_4$};\n\\node (B) at (0:0cm) {$B$};\n\\node (Bl) at (180:\\Rc) {$\\hat{B}$};\n\\node (Br) at (0:\\Rc) {$\\hat{B'}$};\n\\draw[->] (Y1) to [auto,swap,\"\\tiny $E_1$\"] (X1);\n\\draw[->] (Y2) to [auto,\"\\tiny $E_2$\"] (X2);\n\\draw[->] (Y3) to [auto,swap,\"\\tiny $E_2$\"] (X3);\n\\draw[->] (Y4) to [auto,\"\\tiny $E_1$\"] (X4);\n\\draw[->] (X1) to (Bl);\n\\draw[->] (X2) to (Bl);\n\\draw[->] (X3) to (Br);\n\\draw[->] (X4) to (Br);\n\\draw[->] (Bl) to (B);\n\\draw[->] (Br) to (B);\n\\draw[dashed,->] (X1) to [bend right=38,auto,swap,\"$\\chi_1$\"] (X2);\n\\draw[dotted,->] (X2) to [bend right=38,auto,swap,\"$\\chi_2$\"] (X3);\n\\draw[dashed,->] (X3) to [bend right=38,auto,swap,\"$\\chi_3$\"] (X4);\n\\draw[dotted,->] (X4) to [bend right=38,auto,swap,\"$\\chi_4$\"] (X1);\n\\draw[dotted,-] (Y1) to [bend right=38] (Y2)\n (Y2) to [bend right=38] (Y3)\n (Y3) to [bend right=38] (Y4)\n (Y4) to [bend right=38] (Y1)\n (Bl) to [bend right=20] (Br);\n\\end{tikzpicture}\\end{aligned}$$\n\n$$\\begin{tikzpicture}[scale=1.1,font=\\small,outer sep=-1pt]\n\\foreach \\x in {1,...,4}{\n\\foreach \\n in {1,2}{\n\\FPeval{\\internal}{clip(2*(\\x-2)+\\n+1)}\n\\ifthenelse{\\internal=0}{\\FPeval{\\internal}{8}}{}\n\\FPeval{\\thetest}{clip(\\x-\\n)}\n\\ifthenelse{\\thetest = 0 \\OR \\thetest = 2}\n{\\def\\exponent{'}}\n{\\def\\exponent{}}\n\\node (Y\\internal) at ({\\n*45+\\x*90-22.5}:\\Rb) {$Y_{\\x}\\exponent$};\n\\node (T\\internal) at ({\\n*45+\\x*90-22.5}:\\Ra) {$T_{\\x}\\exponent$};\n}}\n\\node (B) at (0:0cm) {$B$};\n\\foreach \\x in {1,...,4}{\n\\node (X\\x) at (\\x*90+45:\\Rc) {$X_{\\x}$};\n\\draw[->] (X\\x) to (B);\n}\n\\draw[dotted,-] (T1) to [bend right=15] (T2)\n (T2) to [bend right=15] (T3)\n (T3) to [bend right=15] (T4)\n (T4) to [bend right=15] (T5)\n (T5) to [bend right=15] (T6)\n (T6) to [bend right=15] (T7)\n (T7) to [bend right=15] (T8)\n (T8) to [bend right=15] (T1)\n (Y1) to [bend right=15] (Y2)\n (Y3) to [bend right=15] (Y4)\n (Y5) to [bend right=15] (Y6)\n (Y7) to [bend right=15] (Y8);\n\\draw[dashed,->] (X1) to [bend right=35,swap,\"$\\chi_1$\"] (X2);\n\\draw[dashed,->] (X2) to [bend right=35,swap,\"$\\chi_2$\"] (X3);\n\\draw[dashed,->] (X3) to [bend right=35,swap,\"$\\chi_3$\"] (X4);\n\\draw[dashed,->] (X4) to [bend right=35,swap,\"$\\chi_4$\"] (X1);\n\\draw[->] (Y1) to [pos=0,\"\\tiny $E_1$\"] (X1);\n\\draw[->] (Y2) to [swap,pos=0,\"\\tiny $E_2$\"] (X2);\n\\draw[->] (Y3) to [pos=0,\"\\tiny $F_1$\"] (X2);\n\\draw[->] (Y4) to [swap,pos=0,\"\\tiny $F_2$\"] (X3);\n\\draw[->] (Y5) to [pos=0,\"\\tiny $E_2$\"] (X3);\n\\draw[->] (Y6) to [swap,pos=0,\"\\tiny $E_1$\"] (X4);\n\\draw[->] (Y7) to [pos=0,\"\\tiny $F_2$\"] (X4);\n\\draw[->] (Y8) to [swap,pos=0,\"\\tiny $F_1$\"] (X1);\n\\draw[->] (T1) to [pos=0.2,\"\\tiny $F_1$\"] (Y1);\n\\draw[->] (T2) to [swap,pos=0.2,\"\\tiny $F_1$\"] (Y2);\n\\draw[->] (T3) to [pos=0.2,\"\\tiny $E_2$\"] (Y3);\n\\draw[->] (T4) to [swap,pos=0.2,\"\\tiny $E_2$\"] (Y4);\n\\draw[->] (T5) to [pos=0.2,\"\\tiny $F_2$\"] (Y5);\n\\draw[->] (T6) to [swap,pos=0.2,\"\\tiny $F_2$\"] (Y6);\n\\draw[->] (T7) to [pos=0.2,\"\\tiny $E_1$\"] (Y7);\n\\draw[->] (T8) to [swap,pos=0.2,\"\\tiny $E_1$\"] (Y8);\n\\end{tikzpicture}$$\n\nThe Sarkisov link $\\chi_1$ is given by a diagram $$\\begin{tikzcd}[link]\n Y_1\\ar[dd,\"\\pi_1\",swap]\\ar[rr,dotted,-]&& Y_2\\ar[dd,\"\\pi_2\"] \\\\ \\\\\nX_1\\ar[rr,\"\\chi_1\",dashed,->] \\ar[rd]&& X_2 \\ar[ld]\\\\\n&\\hat{B}\n\\end{tikzcd}$$ where $X_1,X_2,Y_1,Y_2$ are varieties of dimension $n$, and $\\dim \\hat{B} = n-1$. Denote by $E_1\\subset Y_1$ and $E_2\\subset Y_2$ the respective exceptional divisors of the divisorial contractions $\\pi_1$ and $\\pi_2$. We denote again by $E_1,E_2\\subset T$ the strict transforms of these divisors, under the birational contractions $T\\rat Y_1$ and $T\\rat Y_2$. Then by Lemma \\[lem:SarkiIIConic\\]\\[SarkiII:4\\], $E_1 \\cup E_2$ is a pair of divisors of type for $Y_1/\\hat{B}$, hence also for $T/B$ by Lemma \\[lem:gonalityOfTypeII\\]. ", "By Proposition \\[pro:rankrFibrations\\]\\[rankr:4\\], we are in one of the following mutually exclusive three cases:\n\n1. ", " \\[BisnotBhat\\] $B$ is $\\Q$-factorial, and there exists a divisor $G$ of type for $T/B$.\n\n2. ", " \\[BisnotBhat\\_bis\\] $B$ is not $\\Q$-factorial.", "\n\n3. ", " \\[BisBhat\\] $B$ is $\\Q$-factorial, and there exists another pair $F_1 \\cup F_2$ of divisors of type for $T/B$.\n\nWe denote $\\{X_i/B_i\\}$ the finite collection of all rank $1$ fibrations dominated by $T/B$. In each case we are going to show that this collection has cardinal 4.", "\n\nSuppose first that \\[BisnotBhat\\] holds. ", "By Proposition \\[pro:rankrFibrations\\]\\[rankr:0\\]\\[rankr:3\\] and Lemma \\[lem:factorThroughRank1\\], we can obtain such an $X_i/B_i$ by a birational contraction contracting one the following four sets of divisors: $\\{E_1\\}$, $\\{E_2\\}$, $\\{E_1,G\\}$ or $\\{E_2,G\\}$. Moreover $X_i/B_i$ is determined up to isomorphism by such a choice of contracted divisors:\n\n- If $T \\rat X_i$ contracts $\\{E_1,G\\}$ or $\\{E_2,G\\}$, then $\\rho(X_i) = \\rho(T) - 2$ which implies $\\rho(B_i/B) = 0$, that is, $B_i \\to B$ is an isomorphism. ", "Then $X_i$ is uniquely determined by Corollary \\[cor:corti3.5bis\\].", "\n\n- If $T \\rat X_i$ contracts $\\{E_1\\}$ or $\\{E_2\\}$, then $\\rho(B_i/B) = 1$, and $B_i \\rat B$ is a birational contraction contracting the image of the divisor $G$. Then such a $B_i$ is uniquely determined by Corollary \\[cor:SameDivisors\\].", "\n\nIn conclusion the relation given by Proposition \\[pro:from T3\\] has the form $$\\chi_4\\circ \\chi_3\\circ \\chi_2\\circ \\chi_1=\\id,$$ and more precisely, up to a cyclic permutation exchanging the role of $\\chi_1$ and $\\chi_3$, we have a commutative diagram as in Figure \\[fig:relations\\], top left, where $\\chi_2$ and $\\chi_4$ have respectively type and , and $\\chi_1$ and $\\chi_3$ are equivalent Sarkisov links of type II.", "\n\nNow consider Case \\[BisnotBhat\\_bis\\]. ", "As $\\hat B$ is $\\Q$-factorial (Proposition \\[pro:sing\\_of\\_B\\]), we have $\\hat B \\neq B$, hence $\\rho(\\hat B/B) = 1$ and the morphism $\\hat B\\to B$ is a small contraction. ", "By uniqueness of log-flip, there are exactly two small contractions from a $\\Q$-factorial variety to $B$. Denote $\\hat B' \\to B$ the other one. ", "Then for each $X_i/B_i$, we have $B_i \\simeq \\hat B$ or $\\hat B'$, and $\\rho(X_i/B) = 2$. Hence the birational contraction $T \\rat X_i$ contracts exactly one divisor, which must be $E_1$ or $E_2$. Again this gives four possibilities. ", "The actual existence of $X_3/ \\hat B'$ and $X_4/ \\hat B'$ arises from the two-rays games $X_1/B$ and $X_2/B$. We get a relation as in Figure \\[fig:relations\\], top right, with $\\chi_1, \\chi_3$ of type and $\\chi_2, \\chi_4$ of type .", "\n\nFinally consider Case \\[BisBhat\\]. ", "Then by Proposition \\[pro:rankrFibrations\\]\\[rankr:0\\]\\[rankr:3\\], each birational contraction $T \\rat X_i$ contracts one divisor among $E_1 \\cup E_2$, and another one among $F_1 \\cup F_2$. Again this gives four possibilities. ", "In each case $\\rho(B_i/B) = 0$ hence $B_i$ is isomorphic to $B$, and then Corollary \\[cor:corti3.5bis\\] says that $X_i$ is determined up to isomorphism by such a choice. ", "We obtain a relation with four links of type , as on Figure \\[fig:relations\\], bottom.", "\n\nExample \\[ex:X3X2X1=1\\] illustrates why the assumption on the covering gonality is necessary in Proposition \\[pro:X4X3X2X1=id\\].", "\n\nProof of Theorem \\[TheoremBirMori\\] {#sec:proofBirMori}\n-----------------------------------\n\nIn order to prove Theorem \\[TheoremBirMori\\], we use the generators and relations of $\\BirMori(X)$ which are given in Theorem \\[thm:sarkisov\\]. ", "The key results are then Propositions \\[pro:boundOnGenus\\] and \\[pro:X4X3X2X1=id\\].", "\n\nWe choose the integer $d$ associated to the dimension $n$ by Proposition \\[pro:boundOnGenus\\], and set $M = \\max\\{d, 8\\conngon(X)\\}$. By Theorem \\[thm:sarkisov\\]\\[sarkisov1\\], the groupoid $\\BirMori(X)$ is generated by Sarkisov links and automorphisms of Mori fibre spaces. ", "By Theorem \\[thm:sarkisov\\]\\[sarkisov2\\], the relations are generated by elementary relations, so it suffices to show that every elementary relation is sent to the neutral element in the group $$\\bigast_{C\\in \\CB(X)} \\left(\\bigoplus_{\\MC(C)}\\Z/2\\right).$$\n\nLet $\\chi_t\\circ\\cdots\\circ\\chi_1=\\id$ be an elementary relation, coming from a rank $3$ fibration $T/B$. We may assume that one of the $\\chi_i$ is a Sarkisov link of conic bundles of type with $\\covgon(\\chi_i) > M$, otherwise the relation is sent onto the neutral element as all $\\chi_i$ are sent to the neutral element. ", "We may moreover conjugate the relation and assume that $\\chi_1$ is a Sarkisov link of conic bundles of type with $\\covgon(\\chi_1)>M$. By Proposition \\[pro:boundOnGenus\\], we have $\\dim(B)=n-1$. Then, Proposition \\[pro:X4X3X2X1=id\\] implies that $t=4$ and that $\\chi_1$ and $\\chi_3$ are equivalent Sarkisov links of conic bundles of type . ", "Applying the same argument to the relation $\\chi_1\\circ \\chi_4\\circ \\chi_3\\circ \\chi_2=\\id$ we either find that both $\\chi_2$ and $\\chi_4$ are sent to the neutral element or are equivalent Sarkisov links of conic bundles of type (again by Proposition \\[pro:X4X3X2X1=id\\]). ", "Moreover, all the conic bundles involved in this relation are equivalent. ", "This proves the existence of the groupoid homomorphism.", "\n\nThe fact that it restricts to a group homomorphism from $\\Bir(X)$ is immediate, and the fact that it restricts as a group homomorphism $$\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$$ follows from Lemma \\[lem:SarkisovOverBase\\].", "\n\nImage of the group homomorphism given by Theorem \\[TheoremBirMori\\] {#sec:surjectivity}\n===================================================================\n\nIn this section, we study the image of $\\Bir(X)$ under the group homomorphism given by Theorem \\[TheoremBirMori\\], and more precisely the image of $\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$ for some conic bundles $X/B$. To simplify the notation, we will identify an equivalence class of marked conic bundles in $\\MC(X/B)$ with the associated generator of $\\Z/2$. We can then speak about sums of elements of $\\MC(X/B)$, which we see in $\\bigoplus_{\\MC(X/B)} \\Z/2$, twice the same class being equal to zero.", "\n\nA criterion {#sec:criterion}\n-----------\n\n\\[def:ParityPhiGamma\\] Let $(X/B,\\Gamma)$ be a marked conic bundle, and $\\phi\\colon X/B \\rat Y/B$ a birational map over $B$ between conic bundles. ", "For a general point $p\\in \\Gamma$, and an irreducible curve $C\\subseteq B$ transverse to $\\Gamma$ at $p$, let $b \\in \\N$ be the number of base-points of the birational surface map $\\eta_X^{-1}(C)\\rat \\eta_Y^{-1}(C)$ induced by $\\phi$ that are equal or infinitely near to a point of the fibre of $p$. We call the class $\\bar b \\in \\Z/2$ *the parity of $\\phi$ along $\\Gamma$*.", "\n\nThe following lemma shows that this definition does not depend on the choice of $p$ or $C$. ", "We shall use it to compute the image of the group homomorphism of Theorem \\[TheoremBirMori\\] by studying locally a birational map near a hypersurface $\\Gamma$ of the base.", "\n\n\\[lem:SequenceParity\\] Let $\\eta_X\\colon X\\to B$ and $\\eta_Y\\colon Y\\to B$ be two conic bundles, $\\phi\\colon X\\rat Y$ a birational map over $B$, and $\\Gamma\\subset B$ an irreducible hypersurface not contained in the discriminant locus of $X/B$.\n\nFor any decomposition $\\phi=\\chi_t\\circ \\cdots\\circ \\chi_1$ as in Lemma $\\ref{lem:SarkisovOverBase}$, the parity of $\\phi$ along $\\Gamma$ is equal to the parity of the number of indexes $i\\in \\{1,\\dots,t\\}$ such that $\\chi_i$ is a Sarkisov link of type whose marking $\\Gamma_i\\subset B_i$ is sent to $\\Gamma$ via $B_i/B$.\n\nFix a decomposition $\\phi=\\chi_t\\circ \\cdots\\circ \\chi_1$ as in Lemma $\\ref{lem:SarkisovOverBase}$, a general point $p\\in \\Gamma$ and an irreducible curve $C\\subseteq B$ transverse to $\\Gamma$ at $p$. In particular $p$ is a smooth point of both $\\Gamma$ and $C$. ", "For $i=0,\\dots,t$, we denote by $\\eta_i\\colon X_i\\to B$ the morphism given by the composition $X_i\\to B_i\\to B$.\n\nIt suffices to prove, for $i=0,\\dots,t$, that the following holds:\n\n1. ", " \\[parity:a\\] The morphism $\\eta_i^{-1}(C)\\to C$ has general fibre $\\p^1$, and the fibre over $p$ is $\\p^1$ (this means that $\\Gamma$ is not in the discriminant locus).", "\n\n2. ", " \\[parity:b\\] If $i\\ge 1$, then $\\chi_i\\circ \\cdots\\circ \\chi_1$ induces a birational map between surfaces over $C$ $$\\eta_0^{-1}(C)=\\eta_X^{-1}(C)\\rat \\eta_i^{-1}(C)$$ and the number of base-points that are equal or infinitely near to a point of the fibre of $p$ has the same parity as the number of integers $j\\in \\{1,\\dots,i\\}$ such that $\\chi_j$ is a Sarkisov link of type with marking $\\Gamma_j\\subset B_j$, sent to $\\Gamma$ via $B_j/B$.\n\nWe proceed by induction on $i$. If $i=0$, \\[parity:a\\] follows from the assumption that $\\Gamma$ is not contained in the discriminant locus of $X/B$, and \\[parity:b\\] is clear.", "\n\nFor $i\\ge1$, the birational map $\\chi_i$ induces a birational map over $C$ $$\\theta_i\\colon \\eta_{i-1}^{-1}(C)\\rat \\eta_i^{-1}(C).$$ If $\\chi_i$ is a Sarkisov link of type with marking $\\Gamma_i\\subset B_i$, sent to $\\Gamma$ via $B_i/B$, it follows from the description of $\\chi_i$ given in Lemma \\[lem:SarkiIIConic\\] that the restriction $\\theta_i$ is the composition of the blow-up of a point on the fibre of $p$, the contraction of the strict transform of the fibre and of a birational map that is an isomorphism over an open subset of $C$ that contains the fibre of $p$. This achieves the proof of \\[parity:a\\] and \\[parity:b\\] in this case, using the induction hypothesis.", "\n\nIf $\\chi_i$ is a Sarkisov link of type with a marking not sent to $\\Gamma$ or if $\\chi_i$ is a Sarkisov link of type or , then the restriction $\\theta_i$ of $\\chi_i$ is an isomorphism over an open subset of $C$ that contains the fibre of $p$. This follows again from Lemma \\[lem:SarkiIIConic\\] if the Sarkisov link is of type and from Corollary \\[cor:SarkiIConic\\] if it is of type or . ", "As before, this gives the result by applying the induction hypothesis.", "\n\n\\[cor:Parity\\] Let $X/B$ be a conic bundle, where $\\dim(X)\\ge 3$, and $\\phi\\in \\Bir(X/B)$. The image of $\\phi$ under the group homomorphism $$\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$$ of Theorem \\[TheoremBirMori\\] is equal to the sum of equivalence classes of marked conic bundles $(X/B,\\Gamma)$ with $\\covgon(\\Gamma) > \\max\\{ d, 8\\conngon(X) \\}$ such that the parity of $\\phi$ along $\\Gamma$ is odd.", "\n\nSet $M=\\max\\{ d, 8\\conngon(X) \\}$. Using Lemma \\[lem:SarkisovOverBase\\], we decompose $\\phi$ as $\\phi=\\chi_t\\circ \\cdots \\circ\\chi_1$ where each $\\chi_i$ is a Sarkisov link of conic bundles from $X_{i-1}/B_{i-1}$ to $X_i/B_i$. Denote by $J\\subseteq\\{1,\\dots,t\\}$ the subset of indexes $i$ such that the Sarkisov link $\\chi_i$ is of type and satisfies $\\covgon(\\chi_i) >M$. By definition of the group homomorphism $$\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$$ of Theorem \\[TheoremBirMori\\], the image of $\\phi$ is the sum of the equivalence classes of marked conic bundles of $\\chi_i$ where $i$ runs over $J$. For each $i\\in J$, the marked conic bundle of $\\chi_i$ is equal to $(X_i/B_i,\\hat\\Gamma_i)$ for some irreducible hypersurface $\\hat\\Gamma_i\\subset B_i$ with $\\covgon(\\hat\\Gamma_i) >M$. Hence, $(X_i/B_i,\\hat\\Gamma_i)$ is equivalent to $(X/B,\\Gamma_i)$, where $\\Gamma_i\\subset B$ is the image of $\\hat\\Gamma_i\\subset B_i$ via $B_i/B$. This implies that the image of $\\phi$ is the sum of the classes of $(X/B,\\Gamma_i)$, where $i$ runs over $J$.\n\nBy Lemma \\[lem:SequenceParity\\], this sum is equal to the sum of equivalence classes of marked conic bundles $ (X/B,\\Gamma)$ with $\\covgon(\\Gamma)>M$ and such that the parity of $\\phi$ along $\\Gamma$ is odd.", "\n\nThe case of trivial conic bundles and the proof of Theorem \\[TheoremBirPnnotsimple\\] {#sec:trivialBundle}\n------------------------------------------------------------------------------------\n\nGiven a variety $B$, let $X=\\p^1\\times B$, and $X/B$ the second projection. ", "The group $\\Bir(X/B)$ is canonically isomorphic to $\\PGL_2(\\C(B))$, via the action $$\\begin{tikzcd}[map]\n\\PGL_2(\\C(B)) \\times X & \\rat &X\\\\\n \\left(\\begin{pmatrix} a(t) & b(t)\\\\ c(t) & d(t) \\end{pmatrix} , ([u:v],t)\\right) & \\mapsrat &([a(t)u+b(t)v:c(t)u+d(t)v],t).", "\n\\end{tikzcd}$$\n\nFor $B=\\p^{n-1}$, the group $\\Bir(X/B)$ corresponds, via a birational map $X\\dasharrow \\p^n$ sending the fibres of $X/B$ to lines through a point $p\\in \\p^n$, to the subgroup of the Jonquières group associated to $p$ consisting of birational maps of $\\p^n$ that preserves a general line through $p$ (in general a Jonquières element permutes such lines). ", "Hence, $\\Bir(X/B)$ corresponds to the factor $\\PGL_2(\\C(x_2, \\dots, x_n))$ of the group $\\PGL_2(\\C(x_2, \\dots, x_n)) \\rtimes \\Bir(\\p^{n-1}) \\subseteq \\Bir(\\p^n)$ described in §\\[sec:Generators\\].", "\n\nFor $B$ general, we obtain many different varieties $X=\\p^1\\times B$. It can also be that $X$ is rational even if $B$ is not, but then the conic bundle $X/B$ is not equivalent to the trivial one $\\p^n\\times \\p^1/\\p^n$.\n\n\\[lem:PGL2toG\\] Any surjective group homomorphism $\\tau\\colon \\PGL_2(\\C(B)) \\tto G$ that is not an isomorphism factorises through the quotient $$\\PGL_2(\\C(B))/\\PSL_2(\\C(B)) \\simeq \\C(B)^*/(\\C(B)^*)^2,$$ where the isomorphism corresponds to the determinant. ", "In particular the target group $G$ is abelian of exponent $2$.\n\nThere exists a non trivial element $M \\in \\Ker \\tau$ by assumption. ", "Since the group $\\PGL_2(\\C(B))$ has trivial centre, we can find $N \\in \\PGL_2(\\C(B))$ that does not commute with $M$. Then $\\id\\neq MNM^{-1}N^{-1} \\in \\PSL_2(\\C(B)) \\cap \\Ker \\tau$, and since $\\PSL_2(\\C(B))$ is a simple group we get $\\PSL_2(\\C(B)) \\subseteq \\Ker \\tau$, which gives the result.", "\n\nWrite $\\div \\colon \\C(B)^*\\to \\Div(B)$ the classical group homomorphism that sends a rational function onto its divisor of poles and zeros, and whose image is the group of principal divisors on $B$. Denoting by $\\Pl_B$ the set of prime divisors on $B$, the group homomorphism $\\div$ naturally gives a group homomorphism $$\\PGL_2(\\C(B))/\\PSL_2(\\C(B)) \\simeq\\C(B)^*/(\\C(B)^*)^2\\to \\bigoplus_{\\Pl_B} \\Z/2.$$ We project onto the sum of prime divisors with large enough covering gonality and identify the ones which are equivalent up to a birational map of $B$. This identification corresponds to taking orbits of the action of $\\Aut_\\C(\\C(B))$ on $\\C(B)$. The following lemma shows that the resulting group homomorphism extends from $\\Bir(X/B)$ to $\\Bir(X)$, and coincides with the group homomorphism from Theorem \\[TheoremBirMori\\].", "\n\nObserve that for each $M\\in\\PGL_2(\\C(B))$, we can speak about the parity of the multiplicity of $\\det(M)\\in \\C(B)^*/(\\C(B)^*)^2$ as pole or zero along an irreducible hypersurface $\\Gamma\\subset B$, as the multiplicity of an element of $(\\C(B)^*)^2$ is always even.", "\n\n\\[lem:ImagePGL2\\] Let $B$ be a smooth variety of dimension at least $2$, $X=\\p^1\\times B$, and let $\\phi_M\\in \\Bir(X/B)\\simeq \\PGL_2(\\C(B))$ be the birational map $$\\phi_M\\colon ([u:v],t)\\mapsrat ([a(t)u+b(t)v:c(t)u+d(t)v],t),$$ where $$M=\\begin{pmatrix} a(t) & b(t)\\\\ c(t) & d(t) \\end{pmatrix} \\in \\PGL_2(\\C(B)).$$ The image of $\\phi_M$ under the group homomorphism $$\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$$ of Theorem \\[TheoremBirMori\\] is equal to the sum of the equivalence classes of marked conic bundles $(X/B,\\Gamma)$ such that $\\Gamma\\subset B$ is a irreducible hypersurfaces of $B$ with $\\covgon(\\Gamma) > \\max\\{ d, 8\\conngon(X) \\}$ and such that the multiplicity of $\\det(M)$ along $\\Gamma$ is odd.", "\n\nWe first observe that the image of $\\PSL_2(\\C(B))\\subseteq \\PGL_2(\\C(B))\\simeq \\Bir(X/B)$ under the group homomorphism $$\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$$ is trivial, since $\\PSL_2(\\C(B))$ is simple and not abelian. ", "Hence, the image of an element $\\phi\\in \\Bir(X/B)\\simeq\\PGL_2(\\C(B))$ is uniquely determined by its determinant $\\delta\\in \\C(B)^*/(\\C(B)^*)^2$ (Lemma \\[lem:PGL2toG\\]), and is the same as the image of the dilatation $$\\psi_\\delta\\colon ([u:v],t)\\mapsto ([\\delta(t)u:v],t).$$ So we only need to prove the result for $M$ equal to such a dilatation.", "\n\nWe denote as before by $\\Pl_B$ the set of prime divisors on $B$. For $\\delta\\in \\C(B)^*$ and $\\Gamma\\in \\Pl_B$, we denote by $m_\\delta(\\Gamma)\\in \\Z$ the multiplicity of $\\delta$ along $\\Gamma$, so that $$\\div(\\delta)=\\sum_{\\Gamma\\in \\Pl_B} m_\\delta(\\Gamma)\\, \\Gamma.$$ We also denote by $P_\\delta(\\Gamma)\\in \\{0,1\\}$ the parity of $\\psi_\\delta$ along $\\Gamma$ as defined in Definition \\[def:ParityPhiGamma\\] and Lemma $\\ref{lem:SequenceParity}$. The image of the dilatation $\\psi_\\delta\\in \\Bir(X/B)$ under the group homomorphism $$\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$$ is equal to the sum of equivalence classes of marked conic bundles $(X/B,\\Gamma)$ such that $\\Gamma\\subset B$ is an irreducible hypersurface with $$\\covgon(\\Gamma) > \\max\\{ d, 8\\conngon(X) \\}$$ and such that $P_\\delta(\\Gamma)$ is odd (Corollary \\[cor:Parity\\]). ", "To prove the result, it suffices to show that $P_\\delta(\\Gamma)$ and $m_\\delta(\\Gamma)$ have the same parity. ", "For all $\\delta,\\delta'\\in \\C(B)^*$, we have $$m_\\delta(\\Gamma)+m_{\\delta'}(\\Gamma)=m_{\\delta\\cdot \\delta'}(\\Gamma)\n\\quad\\text{ and }\\quad\nP_\\delta(\\Gamma)+P_{\\delta'}(\\Gamma) \\equiv P_{\\delta\\cdot \\delta'}(\\Gamma)\\pmod{2}.$$ Indeed, the first equality follows from the definition of the multiplicity and the second follows from Lemma $\\ref{lem:SequenceParity}$, since $\\psi_\\delta\\circ \\psi_{\\delta'}=\\psi_{\\delta\\cdot \\delta'}$. The local ring $\\Ol_\\Gamma(B)$ being a DVR, the group $\\C(B)^*$ is generated by elements $\\delta\\in \\C(B)^*$ with $m_\\delta(\\Gamma)=0$, and by one single element $\\delta_0$ which satisfies $m_{\\delta_0}(\\Gamma)=1$. It therefore suffices to consider the case where $m_{\\delta}(\\Gamma)\\in \\{0,1\\}$.\n\nWe take a general point $p\\in \\Gamma$, an irreducible curve $C\\subseteq B$ transverse to $\\Gamma$ at $p$, and compute the number of base-points of the birational map $\\theta\\colon \\p^1\\times C \\rat \\p^1\\times C$ given by $([u:v],t)\\mapsto ([\\delta(t)u:v],t)$ that are equal or infinitely near to a point of the fibre of $p$. If $m_{\\delta}(\\Gamma)=0$, then $\\delta$ is well defined on $p$, so the birational map $\\theta$ induces an isomorphism $\\p^1 \\times\\{ p\\} \\to \\p^1\\times \\{p\\}$, which implies that $P_\\delta(\\Gamma)=0$. If $m_{\\delta}(\\Gamma)=1$, then $\\delta$ has a zero of multiplicity one at $p$, so $\\theta$ has exactly one base-point on $\\p^1\\times \\{p\\}$, namely $([1:0],p)$. The composition of $\\theta$ with the blow-up of $Z\\to \\p^1 \\times C$ of $([1:0],p)$ yields a birational map $Z\\rat \\p^1\\times C$ with no more base-point on the exceptional divisor, as the multiplicity of both $\\delta$ and $v/u$ at the point is $1$, so $P_\\delta(\\Gamma)=1$. This achieves the proof.", "\n\nWe can now give the proof of Theorem \\[TheoremBirPnnotsimple\\].", "\n\nWe denote by $\\Dil_\\k$ the subgroup of birational dilatations $$\\begin{aligned}\n\\Dil_\\k &= \\{(x_1,\\dots,x_n)\\mapsrat \\left(x_1\\alpha(x_2,\\cdots,x_n),x_2,\\dots,x_n\\right)\\mid\n\\alpha\\in\\k(x_2,\\dots,x_{n})^*\\} \\\\\n&\\subseteq \\Bir_\\k(\\p^n) \\simeq \\Aut_\\k(\\k(x_1,\\dots,x_n)).\\end{aligned}$$ We denote $B=\\p^{n-1}$ and use the birational map (defined over $\\k$) $$\\begin{tikzcd}[map]\n X=\\p^1\\times B & \\rat & \\p^n\\\\\n( [u:1], [t_1,\\dots,t_{n-1}:1]) & \\mapsrat & {[1:u:t_1:\\cdots:t_{n-1}]}\n\\end{tikzcd}$$ that conjugates $\\Bir(X)$ to $\\Bir(\\p^n)$, sending elements of the form $$\\{([u:v],t)\\mapsrat ([\\alpha(t)u:v],t)\\mid\\alpha\\in \\C(B)^*\\}$$ onto elements locally given by $(x_1,\\dots,x_n)\\mapsrat \\left(x_1\\alpha(x_2,\\cdots,x_n),x_2,\\dots,x_n\\right)$.\n\nNow we pick a large enough integer $D$ and consider the set $\\Hl_D$ of degree $D$ irreducible hypersurfaces in $\\p^{n-1}$. For each element $\\Gamma\\in \\Hl_D$, we consider an irreducible polynomial $P \\in \\k[x_0,\\dots,x_n]$ of degree $D$ defining the hypersurface $\\Gamma$, choose $\\alpha=P/x_0^D \\in \\k(\\p^{n-1})$ and associate to $\\Gamma$ the element $\\phi_{\\alpha}\\in \\Bir(X/B)$ given by $$\\phi_{\\alpha}\\colon ([u:v],t)\\mapsrat ([\\alpha(t)u:v],t).$$ By Lemma \\[lem:ImagePGL2\\], the image of $\\phi_{\\alpha}$ under the group homomorphism $$\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$$ of Theorem \\[TheoremBirMori\\] is the unique marked conic bundle $(X/B,\\Gamma)$ (as the hypersurface $\\Gamma_0\\subset B$ given by $x_0=0$ satisfies $\\covgon(\\Gamma_0)=1$). ", "It remains to observe that we have enough elements in $\\Hl_D$, up to birational maps of $\\p^{n-1}$, namely as much as in the field $\\k$. Indeed, if we take two general hypersurfaces $\\Gamma_1,\\Gamma_2\\subset \\p^{n-1}$ of degree $\\ge n+1$, then every birational map $\\Gamma_1\\rat \\Gamma_2$ extends to a linear automorphism of $\\p^{n-1}$; this can be shown by taking the suitable Veronese embedding of $\\p^{n-1}$ such that the canonical divisors of $\\Gamma_1$ and $\\Gamma_2$ become hyperplane sections. ", "The dimension of $\\PGL_{n}(\\k)$ being bounded, for a large enough degree $D$ we obtain a quotient of $\\Hl_D$ by $\\PGL_{n}(\\k)$ which has positive dimension, hence which has the same cardinality as the ground field $\\k$. This quotient can be taken as the indexing set $I$ in the statement of Theorem \\[TheoremBirPnnotsimple\\].", "\n\n\\[rem:thmAandB\\]\n\n1. ", " As all birational dilatations in Theorem \\[TheoremBirPnnotsimple\\] belong to the Jonquières subgroup of elements preserving a pencil of lines, the restriction of the group homomorphism $\\Bir(\\p^n)\\to \\bigoplus_I \\Z/2$ to the Jonquières subgroup also is surjective. ", "We will need other conic bundle structures on rational varieties to obtain Theorem \\[TheoremTame\\].", "\n\n2. ", " The proof of Theorem \\[TheoremBirPnnotsimple\\] uses Lemma \\[lem:ImagePGL2\\] in the case where $B=\\p^{n-1}$. For a general basis $B$ we can prove along the same lines that the image of the subgroup of $\\Bir(X/B)$ given by $$\\{([u:v],t)\\mapsrat ([\\delta(t)u:v],t)\\mid\\delta\\in \\C(B)^*\\}$$ under the group homomorphism $\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$ of Theorem \\[TheoremBirMori\\] is infinite. ", "We omit the proof here, as it is similar to the case of $B=\\p^{n-1}$, and moreover we will prove a more general result in Proposition \\[pro:largeimageAnyCB\\].", "\n\nThe case of non-trivial conic bundles and the proof of Theorem \\[TheoremBirXBnotsimple\\] {#sec:nontrivialBundle}\n----------------------------------------------------------------------------------------\n\nRecall that given a smooth conic $C \\subset \\p^2$ and a point $p \\in \\p^2 \\setminus C$, there is an involution $\\iota(p,C) \\in \\Bir(\\p^2)$ that preserves the conic $C$. ", "It is defined on each general line $L$ through $p$ as the involution that fixes $p$ and exchange the two intersection points $L \\cap C$. ", "We say that $\\iota(p,C)$ is the *involution induced by the projection from $p$*. ", "We now use this construction in family to produce interesting involutions on some conic bundles.", "\n\n\\[lem:CBInvolution\\] Let $B$ be a smooth variety, $\\hat{\\eta}\\colon P\\to B$ a locally trivial $\\p^2$-bundle, and $X\\subset P$ a closed subvariety such that the restriction of $\\hat{\\eta}$ is a conic bundle $\\eta\\colon X\\to B$. Let $s\\colon B\\rat P$ be a rational section $($i.e. a rational map, birational to its image, such that $\\hat{\\eta}\\circ s=\\id_B)$, whose image is not contained in $X$. Let $\\iota\\in \\Bir(X/B)$ be the birational involution whose restriction to a general fibre $\\eta^{-1}(b)$ is the involution induced by the projection from $s(b)$. Let $\\Gamma\\subset B$ be an irreducible hypersurface not contained in the discriminant locus of $\\eta$.\n\nThen the parity of $\\iota$ along $\\Gamma$ $($in the sense of Definition $\\ref{def:ParityPhiGamma})$ is the parity of the multiplicity of $F(s)$ along $\\Gamma$, where $F$ is the local equation of $X$ in $P$.\n\nWe choose a dense open subset $U \\subseteq B$ which intersects $\\Gamma$ and trivialises the $\\p^2$-bundle, and view $X$ locally inside of $\\p^2 \\times U$. It is given by $F\\in \\C(B)[x,y,z]$, homogeneous of degree $2$ in $x,y,z$. The fibre of $\\eta \\colon X\\to B$ over a general point of $\\Gamma$ (respectively of $B$) is a smooth conic. ", "The section $s$ corresponds to $[\\alpha:\\beta:\\gamma]$, where $\\alpha,\\beta,\\gamma\\in \\C(B)$ are not all zero and are uniquely determined by $s$, up to multiplication by an element of $\\C(B)^*$.\n\nThe evaluation $F(\\alpha,\\beta,\\gamma)\\in\\C(B)$ at $s$ is uniquely determined by $s$ up to multiplication by the square of an element of $\\C(B)^*$. The parity of the multiplicity of $F(\\alpha,\\beta,\\gamma)$ along $\\Gamma$ is then well defined. ", "The statement of the lemma consists in showing that this parity is equal to the parity of $\\iota$ along $\\Gamma$.\n\nBy multiplying by a suitable power of a local equation of $\\Gamma$, we can choose that neither $\\alpha,\\beta,\\gamma$ has a pole along $\\Gamma$ and that not all three vanish on $\\Gamma$. Then, the restriction of $\\alpha,\\beta,\\gamma$ gives an element $(\\bar{\\alpha},\\bar{\\beta},\\bar{\\gamma})\\in\\C(\\Gamma)^3\\setminus \\{0\\}$. There exists a matrix in $\\GL_3(\\C(\\Gamma))$ that sends $(\\bar{\\alpha},\\bar{\\beta},\\bar{\\gamma})$ to $(1,0,0)$. By extending this matrix as an element of $\\GL_3(\\C(B))$, we can assume that $(\\alpha,\\beta,\\gamma)=(1,0,0)$.\n\nWe then write the equation of $X$ as $$F=ax^2+bxy+cxz+dy^2+eyz+fz^2$$ where $a,b,c,d,e,f\\in \\C(B)$ have no pole along $\\Gamma$ and are not all simultaneously zero on $\\Gamma$, and obtain that $$F(\\alpha,\\beta,\\gamma)=a.$$ With these coordinates, the involution $\\iota \\in \\Bir(P/B)$ is given by $$\\iota\\colon [x:y:z]\\mapsto [-(x+\\tfrac{b}{a}y+\\tfrac{c}{a}z):y:z].$$\n\nIf $a$ does not vanish on $\\Gamma$, then $\\iota$ is an isomorphism on a general point of $\\Gamma$, and the multiplicity of $\\iota$ at $\\Gamma$ is equal to zero. ", "This achieves the proof in this case.", "\n\nSuppose that $a$ is zero on $\\Gamma$. It implies that either $b$ or $c$ is not zero on $\\Gamma$. We denote by $E\\subset P$ the preimage of $\\Gamma$. It is an irreducible hypersurface and the general fibre of $E/\\Gamma$ is isomorphic to $\\p^1$ because $\\Gamma$ is not contained in the discriminant of $X/B$. The variety $E$ is contracted by $\\iota$ onto the section of $E\\to \\Gamma$ given by $[x:y:z]=[1:0:0]$. We denote by $\\hat\\iota,\\nu\\in \\Bir(P/B)$ the birational maps locally given by $$\\hat\\iota\\colon[x:y:z]\\mapsto [-(x+by+cz):y:z]\\;\\text{ and }\\;\\nu\\colon[x:y:z]\\mapsto [ax:y:z],$$ and observe that $\\iota=\\nu^{-1}\\circ \\hat\\iota \\circ \\nu$. Moreover, the variety $\\hat{X}=\\nu(X)\\subset P$ is locally given by $$\\hat{F}=x^2+bxy+cxz+a(dy^2+eyz+fz^2).$$ In particular, the preimage of $\\Gamma$ in $\\hat{X}$ is the union of the hypersurfaces $E_1,E_2\\subset\\hat{X}$, locally given by $x=0$ and $x+by+cz=0$ respectively. ", "As $b$ and $c$ are not both zero on $\\Gamma$, we have $E_1\\neq E_2$.\n\nFor $i=1,2$, a general fibre of $E_i\\to \\Gamma$, for $i=1,2$, is isomorphic to $\\p^1$. Moreover, the restriction of $\\nu$ induces a birational map $E\\rat E_1$ over $\\Gamma$, corresponding on a general fibre to the projection from a smooth conic to a line, via a point of the conic. ", "The birational involution $\\hat\\iota$ induces a birational transformation of $\\hat{X}$ whose restriction gives a birational map $E_1\\rat E_2$ over $\\Gamma$. We denote by $r\\ge 1$ the multiplicity of $a$ along $\\Gamma$. As $\\Gamma$ is an irreducible hypersurface of $B$, the local ring $\\Ol_{B,\\Gamma}$ of rational functions of $B$ defined on an open subset of $\\Gamma$ is a DVR. ", "We can find $a_1,\\ldots,a_r\\in \\C(B)$ of multiplicity $1$ along $\\Gamma$ such that $a=a_1\\cdot a_2\\cdots a_r$. This allows us to write $\\nu^{-1}$ as $\\nu^{-1}=\\nu_r\\circ \\cdots \\circ \\nu_1$, where $\\nu_i$ is given by $[x:y:z]\\mapsto [x:a_iy:a_iz]$. We write $X_1=\\nu_1(\\hat{X})$ and write $X_i=\\nu_i(X_{i-1})$ for $i=2,\\dots,r$. In particular, $X_r=X$. The equation of $X_i$ is given by $$F_i=a_1\\cdots a_ix^2+bxy+cxz+a_{i+1}\\cdots a_r (dy^2+ eyz+ fz^2).$$ For $i=1,\\dots,r-1$, the preimage of $\\Gamma$ in $X_i$ is the union of $E_{1,i}$ and $E_{i,2}$ given by $x=0$ and $by+cz=0$. Writing $E_1=E_{1,0}$ and $E_2=E_{2,0}$, we obtain that $\\nu_i$ contracts $E_{2,i-1}$ onto a rational section of $\\Gamma$ contained in $E_{2,i}\\setminus E_{1,i}$, for $i=1,\\dots,r$.\n\nTo compute the parity of $\\iota$ along $\\Gamma$, we denote by $C\\subseteq \\Gamma$ a general irreducible curve passing through a general point $p\\in \\Gamma$, and look at the birational map obtained by the restriction of $\\iota$ to the preimage of $C$. ", "Recalling that $\\iota=\\nu^{-1}\\circ (\\hat\\iota \\circ \\nu)$, observe that the first map $\\hat\\iota\\circ \\nu$ is a local isomorphism at the point $p$, while the map $\\nu^{-1}$ corresponds to a sequence of $r$ elementary links. ", "The parity of $\\iota$ along $\\Gamma$ is then the class of $r$ in $\\Z/2$ as desired.", "\n\n\\[def:DecomposableCB\\] We say that a conic bundle $X/B$ is a *decomposable conic bundle* if $X$ and $B$ are smooth, and if we have closed embeddings $B\\hookrightarrow \\p^m$ and $X\\hookrightarrow P$ where $P$ is a $\\p^2$-bundle over $\\p^m$, which is the projection of a decomposable vector bundle of rank $3$. We moreover ask that the morphism $X/B$ comes from the restriction of the $\\p^2$-bundle $P\\to \\p^m$ and that $X\\subset P$ is locally given by equations of degree $2$ in the $\\p^2$-bundle.", "\n\n\\[pro:largeimageAnyCB\\] For each decomposable conic bundle $\\eta\\colon X\\to B$ with $\\dim B \\ge 2$, there are infinitely many involutions in $\\Bir(X/B)$ which have distinct images via the group homomorphism $ \\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$ of Theorem \\[TheoremBirMori\\]. ", "In particular, the image is infinite.", "\n\nWe can see $B$ as a closed subset $B\\subseteq \\p^m$, and obtain that $X\\subset P$, where $\\hat{\\eta}\\colon P\\to \\p^m$ is the projectivisation of a rank $3$ vector bundle. ", "We can thus write $P=\\p(\\mathcal{O}_{\\p^m}\\oplus \\mathcal{O}_{\\p^m}(a)\\oplus \\mathcal{O}_{\\p^m}(b))$ for some $a,b\\ge 0$ (up to twisting and exchanging the factors). ", "We view $P$ as the quotient of $(\\A^3\\setminus \\{0\\})\\times (\\A^{m+1}\\setminus \\{0\\})$ by $(\\mathbb{G}_m)^2$ via $$\\left((\\lambda,\\mu),(x_0,x_1,x_2,y_0,\\ldots,y_m)\\right)\\mapsto (\\lambda x_0,\\lambda\\mu^{-a} x_1,\\lambda\\mu^{-b} x_2,\\mu y_0,\\ldots,\\mu y_m)$$ and denote by $[x_0\\colon x_1 \\colon x_2 \\semicolon y_0 \\colon \\ldots \\colon y_m]$ the class of $(x_0,x_1,x_2,y_0,\\ldots,y_m)$ (see [@AhmOka Definition 2.3, Remark 2.4] for more details).", "\n\nThen $X$ is equal to the preimage of $B$ cut by the zero locus of an irreducible polynomial $G\\in \\C[x_0,x_1,x_2,y_0,\\ldots,y_m]$, that has degree $2$ in $x_0,x_1,x_2$ (and suitable degree in $y_0,\\ldots,y_m$ so that the polynomial is homogeneous for the above action). ", "For each integer $d\\ge 1$, and for general homogeneous polynomials $$\\begin{aligned}\nu_0,v_0\\in \\C[y_0,\\ldots,y_m]_d,&&\nu_1,v_1\\in \\C[y_0,\\ldots,y_m]_{d+a},&&\nu_2,v_2\\in \\C[y_0,\\ldots,y_m]_{d+b}, \\end{aligned}$$ (the subscript corresponding to the degree), the closed subvariety $\\hat\\Gamma\\subset X$ of codimension $2$ given by $$\\hat\\Gamma=\\Bigl\\{([x_0:x_1:x_2 \\semicolon y_0:\\ldots:y_m])\\in X\\subseteq P \\Bigm| \\sum_{i=0}^2 x_i u_i=\\sum_{i=0}^2 x_i v_i=0\\Bigr\\}$$ is smooth, by Bertini theorem.", "\n\nWe now prove that the projection $X\\to B$ induces a birational morphism from $\\hat\\Gamma$ to its image $\\Gamma\\subset B$, an irreducible hypersurface of $B$. Solving the linear system $\\sum_{i=0}^2 x_i u_i=\\sum_{i=0}^2 x_i v_i=0$ in $x_0,x_1,x_2$, we obtain that the preimage of $[y_0:\\ldots:y_m]$ is $[u_1v_2-u_2v_1:u_2v_0-u_0v_2:u_0v_1-u_1v_0 \\semicolon y_0:\\ldots:y_m],$ so the projection induces a birational morphism from $\\hat\\Gamma$ to the hypersurface $\\Gamma\\subset B$ given by the polynomial $G(P_0,P_1,P_2,y_0,\\ldots,y_m)$, where $P_0,P_1,P_2\\in \\C[y_0,\\dots,y_m]$ are the polynomials $P_0=u_1v_2-u_2v_1$, $P_1=u_2v_0-u_0v_2$ and $P_2=u_0v_1-u_1v_0$.\n\nWe now show that the covering gonality $\\covgon(\\hat\\Gamma)=\\covgon(\\Gamma)$ is large if $d$ is large enough. ", "We denote by $H_i,F_j\\subset P$ the hypersurfaces given respectively by $x_i=0$ and $y_j=0$, and obtain that $\\Pic(P)=\\Z H_i\\bigoplus \\Z F_j$ for all $i\\in \\{0,1,2\\}, j\\in \\{0,\\ldots,m\\}$. The class of all $F_j$ is the same and denoted by $F$ and $H_0\\sim H_1+aF\\sim H_2+bF$. Note that $\\hat\\Gamma$ is a complete intersection in $\\hat{\\eta}^{-1}(B)\\subseteq P$ of 3 hypersurfaces equivalent to $H_0+dF, H_0+dF, 2H_0+d_0F$ for some $d_0\\in \\Z$ (depending on the equation of $X$). ", "The canonical divisor of $P$ being equivalent to $-H_0-H_1-H_2-F_0-\\ldots-F_m=-3H_0-(m+1-a-b)F$, we obtain by adjunction that the canonical divisor of $\\hat\\Gamma$ is the restriction to $\\hat \\Gamma$ of a divisor of $P$ equivalent to $H_0+(2d+d_0-m-1+a+b)F$. The morphism associated to $F$ is simply the projection $\\hat\\Gamma\\to \\p^m$, which is birational onto its image. ", "By Lemma \\[lem:BVApEasy\\]\\[BVAmor\\]-\\[BVAPn\\], the divisor $pF$ satisfies $\\BVA_p$, for each integer $p\\ge 0$, and thus $K_{\\hat\\Gamma}$ satisfies $\\BVA_p$ for $p=2d+d_0-m-1+a+b\\ge 0$ if $d$ is large enough, by Lemma \\[lem:BVApEasy\\]\\[BVAeff\\]. ", "This implies that $\\covgon(\\hat\\Gamma)\\ge p+2$ by Theorem \\[thm:BVAK\\]. ", "By choosing $d$ large enough we obtain that $\\covgon(\\Gamma)=\\covgon(\\hat\\Gamma)$ is large.", "\n\nWe now use the construction in Lemma \\[lem:CBInvolution\\] of the involution $\\iota\\in \\Bir(X/B)$ associated with the $\\p^2$-bundle $P/B$ and the rational section $s\\colon B\\rat P$ given by $$[y_0:\\ldots:y_m] \\mapsrat [u_1v_2-u_2v_1:u_2v_0-u_0v_2:u_0v_1-u_1v_0 \\semicolon y_0:\\ldots:y_m].$$\n\nBy Lemma \\[lem:CBInvolution\\], the parity of $\\iota$ along $\\Gamma$ is one and the parity of $\\iota$ along any other irreducible hypersurface of $B$ is zero (as $\\Gamma$ is the zero locus of $G(s)$ by construction). ", "For a large integer $d$, the image of $\\iota$ under the group homomorphism $$\\Bir(X/B)\\to \\bigoplus_{\\MC(X/B)} \\Z/2$$ of Theorem \\[TheoremBirMori\\] is the equivalence class of $(X/B,\\Gamma)$. Taking larger and larger $d$, we obtain infinitely many involutions in the image of the group homomorphisms, which are distinct and thus generate a group isomorphic to an infinite direct sum of $\\Z/2$, as the covering gonality of the hypersurfaces goes to infinity with $d$.\n\nWe use the group homomorphism $$\\Bir(X) \\to \\bigast_{C\\in \\CB(X)} \\left(\\bigoplus_{\\MC(C)}\\Z/2\\right)$$ of Theorem \\[TheoremBirMori\\]. ", "By assumption, $X/B$ is a decomposable conic bundle (in the sense of Definition \\[def:DecomposableCB\\]). ", "By Proposition \\[pro:largeimageAnyCB\\], the image of $\\Bir(X/B)$ contains a group isomorphic to an infinite direct sum of $\\Z/2$.\n\nTo finish the proof of Theorem \\[TheoremBirXBnotsimple\\], we take a subfield $\\k\\subseteq \\C$ over which $X,B$ and $\\eta$ are defined, and check that the involutions in $\\Bir(X/B)$ that are used to provide the large image are defined over $\\k$. Firstly, the involutions provided by Lemma \\[lem:CBInvolution\\] are defined over $\\k$ as soon as the rational section $s\\colon B\\rat P$ is. ", "Secondly, the construction of Proposition \\[pro:largeimageAnyCB\\] works for general polynomials in $\\C[y_0,\\ldots,y_m]$ of some fixed degrees.", "\n\nSince a dense open subset of an affine space $\\A^n_\\C$ contains infinitely many $\\k$-points for each subfield $\\k\\subseteq \\C$ (follows from the fact that the $\\Q$-points of $\\A^n$ are dense), we can assume that the polynomials, and thus the section, are defined over $\\k$.\n\nNon-equivalent conic bundles {#sec:freeProduct}\n============================\n\nIn this section, we construct infinitely many non-equivalent conic bundles on $\\p^n$, showing that the set $\\CB(\\p^n)$ is infinite for $n\\ge 3$ (by contrast, observe that $\\CB(\\p^2)$ consists of one element). ", "This allows us to prove Theorems \\[TheoremManyCB\\] and \\[TheoremTame\\].", "\n\nStudying the discriminant locus\n-------------------------------\n\nThe main result of this section is Proposition \\[pro:Isogenies\\]. ", "We prove in particular that if two standard conic bundles $X_1/\\p^2$ and $X_2/\\p^2$ are ramified over smooth curves $\\Delta_1$ and $\\Delta_2$ such that the conic bundles $(X_1\\times \\p^n)/(\\p^2\\times \\p^n)$ and $(X_1\\times \\p^n)/(\\p^2\\times \\p^n)$ are equivalent, then there exist surjective morphisms $\\Delta_1\\tto \\Delta_2$ and $\\Delta_2\\tto \\Delta_1$.\n\nThe following notion is called an *embedded conic* in [@sarkisov page 358].", "\n\n\\[def:conicfibration\\] Let $V$ be a smooth quasi-projective variety. ", "An *embedded conic fibration* is a projective morphism $\\eta \\colon X\\to V$ that is the restriction of a locally trivial $\\p^2$-bundle $\\hat{\\eta}\\colon P\\to V$, and such that $X\\subset P$ is a closed subvariety, given locally by an equation of degree $2$. Precisely, for each $p\\in V$, there exists an affine open subset $U\\subseteq V$ containing $p$ such that $\\hat{\\eta}^{-1}(U)$ is isomorphic to $U\\times \\p^2$, and the image of $\\eta^{-1}(U) \\subset U\\times \\p^2$ is a closed subvariety, irreducible over $\\C(U)$, and defined by a polynomial $F\\in \\C[U][x,y,z]$ homogeneous of degree $2$ in the coordinates $x,y,z$.\n\n\\[rem:FlatisConicFib\\] Let $\\eta\\colon X\\to V$ be a flat projective morphism between smooth quasi-projective varieties, with generic fibre an irreducible conic. ", "Then, $\\eta$ is an embedded conic fibration in a natural way. ", "This is done by taking the locally trivial $\\p^2$-bundle $P=\\p(\\eta_*(\\omega_X^{-1}))$ over $V$, where $\\omega_X$ is the canonical line bundle of $X$ (see [@sarkisov §1.5]).", "\n\nThe following definition is equivalent to the one of [@sarkisov Definition 1.4].", "\n\n\\[Defi:StandardCB\\] A *standard conic bundle* is a morphism $\\eta\\colon X\\to B$ which is a conic bundle $($in the sense of Definition $\\ref{def:conicBundle})$, and which is moreover *flat* with $X$ and $B$ smooth. ", "This implies that $\\eta$ is also an *embedded conic fibration* in the $\\p^2$-bundle $\\p(\\eta_*(\\omega_X^{-1}))\\to B$ (see Remark \\[rem:FlatisConicFib\\]).", "\n\nLet us make some comparisons between the above definitions.", "\n\nAn *embedded conic fibration* (Definition \\[def:conicfibration\\]) over a projective base is not necessarily a *conic bundle* (Definition \\[def:conicBundle\\]), as the relative Picard rank can be $>1$. Conversely, a conic bundle $X/B$ is not necessarily an embedded conic fibration, but it is one if the conic bundle is *standard* (Definition \\[Defi:StandardCB\\]) (as explained just above) or *decomposable* (Definition \\[def:DecomposableCB\\]).", "\n\nMoreover, a decomposable conic bundle is not always standard, as some fibres can be equal to $\\p^2$. It is not clear to us if there exist standard conic bundles which are not decomposable.", "\n\n\\[def:MultDiscriminant\\] Let $V$ be a smooth quasi-projective variety and $\\eta\\colon X\\to V$ a flat embedded conic fibration.", "\n\nFor each irreducible closed subset $\\Gamma\\subseteq V$, we define the *multiplicity of the discriminant of $\\eta$ along $\\Gamma$* as follows. ", "We take an open subset $U\\subseteq V$ that intersects $\\Gamma$ and such that $\\eta^{-1}(U)$ is a closed subset of $U\\times \\p^2$, of degree $2$, and consider a symmetric matrix $M\\in \\Mat_{3\\times 3}(\\C(U))$ that defines the equation of $\\eta^{-1}(U)$. We choose $M$ such that all coefficients of $M$ are contained in the local ring $\\mathcal{O}_\\Gamma(U)\\subset \\C(U)$ of rational functions defined on a general point of $\\Gamma$, and such that the residue matrix $\\widebar{M}\\in \\Mat_{3\\times 3}(\\C(\\Gamma))$ is not zero. ", "This is possible as the morphism is flat, and defines $M$ uniquely, up to multiplication by an invertible element of $\\mathcal{O}_\\Gamma(U)$.\n\nNow we define the multiplicity of the discriminant of $\\eta$ along $\\Gamma$ to be the least integer $m\\ge 0$ such that the determinant lies in $\\mathfrak{m}_\\Gamma(U)^m$, where $\\mathfrak{m}_\\Gamma(U)$ is the maximal ideal of $\\mathcal{O}_\\Gamma(U)$, kernel of the ring homomorphism $\\mathcal{O}_\\Gamma(U)\\tto \\C(\\Gamma)$.\n\nWe define the *discriminant divisor of $\\eta$* to be $\\sum m_D D$, where the sum runs over all irreducible hypersurfaces $D\\subset V$ and where $m_D\\in \\N$ is the the multiplicity of the discriminant of $\\eta$ along $D$ as defined above.", "\n\nIf $\\eta\\colon X\\to V$ is moreover a conic bundle, the definition of the discriminant given in Definition \\[def:MultDiscriminant\\] is compatible with the definition of discriminant locus given in Definition \\[def:conicBundle\\]: the discriminant locus is the reduced part of the discriminant divisor of $\\eta$. Moreover, if $\\eta$ is a standard conic bundle, the discriminant divisor is reduced [@sarkisov Corollary 1.9]. ", "The multiplicity of the discriminant divisor along irreducible hypersurfaces of $V$ is always $0$ or $1$ in this case. ", "We will however not only consider hypersurfaces but also closed subvarieties of lower dimension.", "\n\nUsing the local description of the matrix that defines $\\eta$ as a flat embedded conic fibration, one can prove the following:\n\n\\[pro:CBDivisorSingFibres\\] Let $V$ be a smooth quasi-projective variety, let $\\eta\\colon X\\to V$ be a flat embedded conic fibration, such that $X$ is smooth. ", "The discriminant divisor $\\Delta$ of $\\eta$ has the following properties: for each point $p\\in V$, the fibre $f_p=\\eta^{-1}(p)$ is given as follows: $$f_p \\text{ is }\n\\begin{cases}\n\\text{a smooth conic}\\\\ \\text{the union of two distinct lines}\\\\ \\text{a double line}\n\\end{cases}\n\\iff \np\\text{ is }\n\\begin{cases}\n\\text{not on }\\Delta\\\\ \\text{a smooth point of }\\Delta \\\\ \\text{a singular point of }\\Delta.", "\n\\end{cases}$$\n\nWe shall need the following folklore result:\n\n\\[lem:FlatConicFibDisc\\] Let $V$ be a smooth quasi-projective variety and let $\\eta_1\\colon X_1\\to V$ and $\\eta_2\\colon X_2\\to V$ be two flat embedded conic fibrations. ", "Let $\\psi\\colon X_1\\rat X_2$ be a birational map over $V$. Let $\\Gamma\\subseteq V$ be a closed irreducible subvariety such that $\\eta_1^{-1}(\\Gamma)$ is not contained in the base-locus of $\\psi$, that the preimage $\\eta_2^{-1}(\\Gamma)$ is irreducible and that a general fibre of $\\eta_2^{-1}(\\Gamma)\\to \\Gamma$ is the union of two distinct lines. ", "We moreover assume that the multiplicity of the discriminant of $\\eta_2$ along $\\Gamma$ is $1$. Then, one of the following holds:\n\n1. ", " \\[GammaDiscNonRed\\] Every fibre of $\\eta_1^{-1}(\\Gamma)\\to \\Gamma$ is a double line $($non-reduced fibre$)$.\n\n2. ", " \\[GammaDiscIrrpi\\] The preimage $\\eta_1^{-1}(\\Gamma)$ is irreducible and a general fibre of $\\eta_1^{-1}(\\Gamma)\\to \\Gamma$ is the union of two distinct lines.", "\n\nReplacing $V$ by an open subset that intersects $\\Gamma$, we can assume that $X_1$ and $X_2$ are closed subvarieties of $V\\times \\p^2$ given by a polynomial of degree $2$ in the coordinates of $\\p^2$. We denote by $\\mathcal{O}_\\Gamma(V)\\subset \\C(V)$ the subring of rational functions that are defined on a general point of $\\Gamma$ and consider the surjective residue homomorphism $\\mathcal{O}_\\Gamma(V)\\tto \\C(\\Gamma)$. The quadratic equations of $X_1$ and $X_2$ correspond to symmetric matrices $M_1,M_2\\in \\Mat_{3\\times 3}(\\C(V))$, defined up to scalar multiplication. ", "Since both $\\eta_1$ and $\\eta_2$ are flat, we can choose $M_1,M_2\\in \\Mat_{3\\times 3}(\\mathcal{O}_\\Gamma(V))$ such that the residue matrices $\\widebar{M}_1,\\widebar{M}_2\\in \\Mat_{3\\times 3}(\\C(\\Gamma))$ are not zero.", "\n\nThe fact that $\\eta_2^{-1}(\\Gamma)$ is irreducible and that a general fibre of $\\eta_2^{-1}(\\Gamma)\\to \\Gamma$ is the union of two distinct lines is equivalent to asking that the quadratic form associated to $M_2$ corresponds to a singular irreducible conic over the field $\\C(\\Gamma)$. It then corresponds to the union of two lines defined over an extension of degree $2$ of $\\C(\\Gamma)$, which intersect into a point defined over $\\C(\\Gamma)$. After a change of coordinates on $X_2\\subset V\\times \\p^2$, applying an element of $\\PGL_3(\\C(V))$ which restricts to an element of $\\PGL_3(\\C(\\Gamma))$, we can assume that the point is $[0:0:1]$ and completing the square we assume that the restriction is given by $F=ax^2+by^2$ for some $a,b\\in \\C(\\Gamma)^*$, where $-\\frac{a}{b}\\in \\C(\\Gamma)^*$ is not a square. ", "This corresponds to saying that $\\widebar M_2$ is equal to the diagonal matrix $\\diag(a,b,0)$.\n\nThe birational map $\\psi$ is given by $$\\left(v, \n\\left[\\begin{smallmatrix} x\\\\ y \\\\ z \\end{smallmatrix}\\right]\n\\right) \\mapsrat \\left(v, A(v)\\cdot\n\\left[\\begin{smallmatrix} x\\\\ y \\\\ z \\end{smallmatrix}\\right] \\right)$$ for some $A\\in \\GL_3(\\C(V))$. This implies that $M_1$ and $\\tr{A}\\cdot M_2\\cdot A$ are collinear in $\\Mat_{3\\times 3}(\\C(V))$. As $\\eta_1^{-1}(\\Gamma)$ is not contained in the base-locus of $\\psi$, we can assume that $A\\in \\Mat_{3\\times 3}(\\mathcal{O}_\\Gamma(V))$ is such that its residue $\\widebar{A}\\in \\Mat_{3\\times 3}(\\C(\\Gamma))$ is not zero. ", "We can moreover choose an element $S\\in \\GL_{3}(\\mathcal{O}_\\Gamma(V))$, with residue $\\widebar{S}\\in \\GL_{3}(\\C(\\Gamma))$, and replace $A$ with $A\\cdot S$. This corresponds to a coordinate change of $\\p^2\\times V$ at the source, which only affects $X_1$ and not $X_2$. We can then reduce to the following possibilities for $\\widebar{A}$, according to the rank of the $2\\times 3$ matrix obtained from the first two rows of $\\widebar{A}$: $$\\begin{aligned}\n\\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ \\mu_1 & \\mu_2 & \\mu_3 \\end{pmatrix}, && \n\\begin{pmatrix} \\alpha & 0 & 0 \\\\ \\beta & 0 & 0 \\\\ \\mu_1 & \\mu_2 & \\mu_3 \\end{pmatrix}, &&\n\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\mu_1 & \\mu_2 & \\mu_3 \\end{pmatrix},\\end{aligned}$$ where $\\alpha,\\beta,\\mu_1,\\mu_2,\\mu_3\\in \\C(\\Gamma)$ and $(\\alpha,\\beta)\\not=(0,0)$.\n\nIn the first case, $\\tr{\\widebar{A}}\\cdot \\widebar{M_2}\\cdot \\widebar{A}=\\widebar{M_2}$, so $\\eta_2^{-1}(\\Gamma)$ has the same properties as $\\eta_1^{-1}(\\Gamma)$, which gives \\[GammaDiscIrrpi\\].", "\n\nThe second case gives $\\tr{\\widebar{A}}\\cdot \\widebar{M_2}\\cdot \\widebar{A}= \\diag(\\alpha^2 a+\\beta^2 b,0,0)$. As $(\\alpha,\\beta)\\not=(0,0)$ and $-\\frac{a}{b}\\in \\C(\\Gamma)^*$ is not a square, we have $\\alpha^2 a+\\beta^2b\\not=0$. The quadratic form associated to this matrix is then a double line, and we obtain \\[GammaDiscNonRed\\].", "\n\nIt remains to study the last case, which yields $\\tr{\\widebar{A}}\\cdot \\widebar{M_2}\\cdot \\widebar{A}=0$. This means that all coefficents of $\\tr{A}\\cdot M_2\\cdot A$ belong to the maximal ideal $\\mathfrak{m}=\\mathfrak{m}_\\Gamma(V)$ of $\\mathcal{O}_\\Gamma(V)$, kernel of the residue homomorphism $\\mathcal{O}_\\Gamma(V)\\tto \\C(\\Gamma)$. Applying $S$ as before, we can assume that $\\mu_1=1$, $\\mu_2=\\mu_3=0$, since the rank of $\\widebar{A}$ is $1$. We write $M_2 = \\diag(a,b,0) + (\\nu_{i,j})_{1 \\le i,j \\le 3}$ where $\\nu_{i,j}\\in \\mathfrak{m}$ for all $i,j$, and obtain $\\det(M_2)\\equiv a\\cdot b\\cdot \\nu_{3,3}\\pmod{\\mathfrak{m}^2}$. As the multiplicity of the discriminant of $\\eta_2$ along $\\Gamma$ is $1$, this implies that $\\nu_{3,3}\\in \\mathfrak{m}\\setminus \\mathfrak{m}^2$. We compute $\\tr{A}\\cdot M_2\\cdot A\\equiv \\diag(\\nu_{3,3},0,0) \\pmod{\\mathfrak{m}^2}$. The quadratic form associated to this matrix is a double line, so again we obtain \\[GammaDiscNonRed\\].", "\n\nWe give two examples to illustrate the need for all the assumptions in Lemma \\[lem:FlatConicFibDisc\\]:\n\nWe work over the affine plane $V=\\A^2$ and consider $$\\begin{aligned}\nX_1&= \\{([x:y:z],(u,v))\\in \\p^2\\times \\A^2 \\mid x^2v+y^2-z^2=0\\},\\\\\nX_2&= \\{([x:y:z],(u,v))\\in \\p^2\\times \\A^2 \\mid x^2v+y^2-u^2z^2=0\\},\\\\\nX_2'&= \\{([x:y:z],(u,v))\\in \\p^2\\times \\A^2 \\mid x^2uv+y^2-z^2=0\\}.\\end{aligned}$$ The projection onto the second factor gives three flat embedded conic fibrations $\\eta_1\\colon X_1\\to \\A^2$, $\\eta_2\\colon X_2\\to \\A^2$, $\\eta_2'\\colon X_2'\\to \\A^2$, with discriminant divisors being respectively given by $v=0$, $u^2v=0$ and $uv=0$. The birational maps of $\\p^2\\times \\A^2$ given by $([x:y:z],(u,v))\\mapsto ([xu:yu:z],(u,v))$ and $([x:y:z],(u,v))\\mapsto ([2x:(u+1)y+(u-1)z:(u-1)y+(u+1)z],(u,v))$ provide two birational maps $\\psi\\colon X_1\\rat X_2$ and $\\psi'\\colon X_1\\rat X_2'$ over $\\A^2$.\n\nChoosing $\\Gamma\\subset \\A^2$ to be the line $\\{u=0\\}$, the result of Lemma \\[lem:FlatConicFibDisc\\] does not hold for $\\psi$ and for $\\psi'$, because a general fibre of $\\eta_1^{-1}(\\Gamma)\\to \\Gamma$ is a smooth conic. ", "In both cases, exactly one hypothesis is not satisfied. ", "Namely, the multiplicity of the discriminant of $\\eta_2$ along $\\Gamma$ is $2$ instead of $1$, and the surface $\\eta_2'^{-1}(\\Gamma)$ is not irreducible.", "\n\nThe idea of the proof of the following statement was given to us by C. Böhnig and H.-C. Graf von Bothmer.", "\n\n\\[pro:Isogenies\\] Let $B$ be a smooth surface, and for $i=1,2$, let $\\eta_i\\colon X_i\\to B$ be a standard conic bundle with discriminant a smooth irreducible curve $\\Delta_i\\subset B$. Assume that there exists a commutative diagram $$\\begin{tikzcd}[link]\nX_1\\times Y \\ar[dd,\"\\eta_1\\times \\id\",swap]\\ar[r,\"\\psi\",dashed]& X_2\\times Y\\ar[dd,\"\\eta_2\\times \\id\"] \\\\ \\\\\nB\\times Y \\ar[r,\"\\theta\",dashed]& B\\times Y\n\\end{tikzcd}$$ where $Y$ is smooth and $\\psi$, $\\theta$ are birational.", "\n\nThen, for a general point $p\\in Y$, the image of $\\Delta_1\\times \\{p\\}$ is contained in $\\Delta_2\\times Y$ and the morphism $\\Delta_1\\to \\Delta_2$ obtained by composing $$\\Delta_1 \\iso \\Delta_1 \\times \\{p\\} \\stackrel{\\theta}{\\rat} \\Delta_2 \\times Y\\stackrel{\\pr_1}{\\longto} \\Delta_2$$ is surjective $($here $\\pr_1\\colon \\Delta_2\\times Y\\to \\Delta_2$ is the first projection$)$.\n\nFor $i=1,2$, the discriminant divisor of $\\eta_i$ is reduced [@sarkisov Corollary 1.9], so consists of $\\Delta_i$. As $\\Delta_i$ is smooth, $\\eta_i^{-1}(p)$ is the union of two distinct lines for each $p\\in \\Delta_i$ (Proposition \\[pro:CBDivisorSingFibres\\]). ", "Since $\\rho(X_i/B_i)=1$, the preimage $\\eta_i^{-1}(\\Delta_i)$ is irreducible. ", "The morphism $(X_i\\times Y)/(B\\times Y)$ is a standard conic bundle whose discriminant divisor is reduced, consisting of the smooth hypersurface $\\Delta_i\\times Y\\subset B\\times Y$.\n\nWe choose a dense open subset $U\\subseteq B\\times Y$ on which $\\theta$ is defined and whose complement is of codimension $2$ (since $B\\times Y$ is smooth). ", "In particular, $U\\cap (\\Delta_1\\times Y)$ is not empty, so $U\\cap (\\Delta_1\\times \\{p\\})\\not=\\emptyset$ for a general point $p\\in Y$. After restricting the open subset, we can moreover assume that $\\eta_1^{-1}(U)$ is a closed subset of $U\\times\\p^2$, given by the quadratic form induced by a matrix $M_1\\in \\GL_3(\\C(U))$. The coefficients of the matrix can moreover be chosen in $\\C(B)\\subseteq \\C(B\\times Y)=\\C(U)$, as the equation of $X_1\\times Y$ in $\\p^2\\times Y$ is locally the equation of $X_1$ in $\\p^2$, independent of $Y$.\n\nWe define $C\\subset B\\times Y$ to be image of $\\Delta_1\\times \\{p\\}$ by $\\theta$, which is a point or an irreducible curve, as $\\Delta_1$ is an irreducible curve. ", "The aim is now to show that $C\\subseteq \\Delta_2\\times Y$ and that $\\pr_1(C)=\\Delta_2$. We choose an open subset $V\\subseteq B\\times Y$ intersecting $C$ such that $\\eta_2^{-1}(V)$ is contained in $ \\p^2\\times V$ and is given by the quadratic form given by a symmetric matrix $M_2\\in \\Mat_{3\\times 3}(\\C(V))$. The morphism $\\eta_2$ being flat, we can choose the coefficients of $M_2$ to be defined on $C$ and such that the residue matrix in $\\widebar{M_2}\\in \\Mat_{3\\times 3}(\\C(C))$ is not zero. ", "The birational map $\\psi$ is locally given by $$\\begin{tikzcd}[map]\nU \\times \\p^2 & \\rat & V \\times \\p^2\\\\\n\\left(u, \\left[\\begin{smallmatrix} x\\\\ y \\\\ z \\end{smallmatrix}\\right]\\right)& \\mapsrat & \\left(\\theta(u), A(u) \\cdot \\left[\\begin{smallmatrix} x\\\\ y \\\\ z \\end{smallmatrix}\\right]\\right)\n\\end{tikzcd}$$ for some $A\\in \\GL_3(\\C(U))$. The explicit form of the map $\\psi$ gives $$\\lambda\\cdot M_1= \\tr{A}\\cdot \\theta^{*}(M_2)\\cdot A$$ where $\\lambda\\in \\C(U)^*$ is a scalar and $\\theta^{*}(M_2)$ is the matrix obtained from $M_2$ by applying to its coefficients the field isomorphism $\\theta^{*}\\colon \\C(V)\\to \\C(U)$. As the rational map $\\theta$ induces a dominant rational map $\\Delta_1\\times \\{p\\}\\rat C$, we have a field homomorphism $\\C(C)\\to \\C(\\Delta_1\\times \\{p\\})\\simeq \\C(\\Delta_1)$, that we denote by $\\bar\\theta^*$. It induces a commutative diagram $$\\begin{tikzcd}[link]\n\\mathcal{O}_{C}(V) \\ar[dd,swap]\\ar[r,\"\\theta^*\"]& \\mathcal{O}_{\\Delta_1\\times \\{p\\}}(U) \\ar[dd] \\\\ \\\\\n\\C(C) \\ar[r,\"\\bar\\theta^*\"]& \\C(\\Delta_1\\times \\{p\\}) \\ar[r,phantom,\"\\simeq\"]&[-4em] \\C(\\Delta_1).", "\n\\end{tikzcd}$$\n\nWe denote by $X'\\subset U\\times \\p^2$ the subvariety given by the quadratic form associated to the matrix $\\theta^{*}(M_2)$. We observe that the coefficients of $\\theta^{*}(M_2)$ are defined over $\\Delta_1\\times \\{p\\}$ and that the residue gives a matrix $\\widebar{\\theta^{*}(M_2)}\\in \\Mat_{3\\times 3}(\\C(\\Delta_1))$ which is obtained by applying the field homomorphism $\\widebar{\\theta^*}$ to the entries of $\\widebar{M_2}\\in \\Mat_{3\\times 3}(C)$. The morphism $\\pr_1\\colon X'\\to U$ is then an embedded conic fibration, which is flat after maybe reducing the open subset $U$ (but still having $U\\cap (\\Delta_1\\times \\{p\\})\\not=\\emptyset$).", "\n\nWe can apply Lemma \\[lem:FlatConicFibDisc\\] to the birational map $X'\\rat X$ given by $$\\left(u, \\left[\\begin{smallmatrix} x\\\\ y \\\\ z \\end{smallmatrix}\\right]\\right) \\mapsrat \\left(u, A(u)^{-1} \\cdot \\left[\\begin{smallmatrix} x\\\\ y \\\\ z \\end{smallmatrix}\\right]\\right)$$ and to $\\Gamma=\\Delta_1\\times \\{p\\}$. Indeed, $(\\eta_1 \\times \\id)^{-1}(\\Delta_1\\times \\{p\\})$ is irreducible as $\\eta_1^{-1}(\\Delta_1)$ is irreducible, and every fibre of $(\\eta_1 \\times \\id)^{-1}(\\Delta_1\\times \\{p\\})\\to \\Delta_1\\times \\{p\\}$ is the union of two distinct lines, as the same holds for $\\eta_1^{-1}(\\Delta_1)\\to \\Delta_1$ by Proposition \\[pro:CBDivisorSingFibres\\]. ", "Lemma \\[lem:FlatConicFibDisc\\] gives two possibilities for the matrix $\\bar\\theta^*(M_2)\\in \\Mat_{3\\times 3}(\\C(\\Delta_1))$: either it is of rank $1$ (case \\[GammaDiscNonRed\\]) or it is of rank $2$, corresponding to a singular irreducible conic (case \\[GammaDiscIrrpi\\]). ", "This gives the same two possibilities for $\\widebar{M_2}\\in \\Mat_{3\\times 3}(C)$ as $\\bar{\\theta}^*$ is a field homomorphism. ", "As the rank of $M_2$ is smaller than $3$, the variety $C$ is in the discriminant of $(X_2\\times Y)/(B\\times Y)$ and is thus contained in $\\Delta_2\\times Y$ as desired. ", "It remains to see that $C$ is not contained in $\\{q\\}\\times Y$ for some point $q$. Indeed, the preimage $(\\eta_2\\times \\id)^{-1}(\\{q\\}\\times Y)$ is isomorphic to $\\eta_2^{-1}(\\{q\\})\\times Y$, which is not irreducible, as $\\eta_2^{-1}(\\{q\\})$ is the union of two lines (again by Proposition \\[pro:CBDivisorSingFibres\\]), but which is reduced.", "\n\nConic bundles associated to smooth cubic curves\n-----------------------------------------------\n\nThe principal result in this section is Proposition \\[pro:EnoughCB\\], which provides a family of conic bundles that we shall use in the next section to prove Theorem \\[TheoremManyCB\\].", "\n\n\\[lem:ThreeConics\\] For each $p=[\\alpha:\\beta]\\in \\p^1$, the set $$\\Sl_p=\\{[x_0:x_1:x_2]\\in \\p^2 \\mid \\alpha x_0^2+\\beta x_1x_2=\\alpha x_1^2+\\beta x_0x_2=\\alpha x_2^2+\\beta x_0x_1=0\\}$$ consists of three points if $\\alpha(\\alpha^3+\\beta^3)=0$ and is empty otherwise.", "\n\nSince $\\Sl_{[0:1]}=\\{[1:0:0],[0:1:0],[0:0:1]\\}$ and $\\Sl_{[1:0]}=\\emptyset$, we may assume that $\\alpha\\in \\C^*$ and $\\beta=1$. If $[x_0:x_1:x_2]\\in \\Sl_p$, then $\\alpha(x_0^3-x_1^3)=x_0( \\alpha x_0^2+ x_1x_2)-x_1( \\alpha x_1^2+ x_0x_2)=0$. The equations being symmetric, we get $x_0^3=x_1^3=x_2^3$. In particular $x_0x_1x_2\\not=0$, so the three equations are equivalent to $$\\alpha=-\\frac{x_1x_2}{x_0^2}=-\\frac{x_0x_1}{x_2^2}=-\\frac{x_0x_2}{x_1^2},$$ which implies that $\\alpha^3=-1$. For the three possible values of $\\alpha$, we observe that $\\Sl_{[\\alpha:1]}=\\{[1:x_1:-\\alpha/x_1]\\mid x_1^3=1\\}$ consists of three points.", "\n\n\\[lem:Xxi\\] For each $\\xi\\in \\C$ such that $\\xi^3\\not=-\\frac{1}{8}$, the hypersurface $X_\\xi\\subset \\p^2\\times \\p^2$ of bidegree $(2,1)$ given by $$X_\\xi=\\Bigl\\{([x_0:x_1:x_2],[y_0:y_1:y_2])\\in \\p^2 \\times \\p^2 \\Bigm| \\sum_{i=0}^2 (x_i^2+ 2\\xi \\frac{x_0x_1x_2}{x_i})y_i=0\\Bigr\\}$$ is smooth, irreducible, rational over $\\Q(\\xi)$, and satisfies $\\rho(X_\\xi)=2$. The second projection gives a standard conic bundle $X_\\xi/\\p^2$. The discriminant curve $\\Delta_\\xi\\subset \\p^2$ is given by $$-\\xi^2(y_0^3+y_1^3+y_2^3)+(2\\xi^3+1)y_0y_1y_2=0$$ and is the union of three lines if $\\xi=0$ or if $\\xi^3=1$, and is a smooth cubic otherwise.", "\n\nTo show that $X_\\xi$ is smooth, irreducible, rational over $\\Q(\\xi)$ and that $\\rho(X_\\xi)=2$, it suffices to show that the first projection $X_\\xi\\to \\p^2$ is a (Zariski locally trivial) $\\p^1$-bundle. ", "This amounts to showing that the coefficients of the linear polynomial in the variables $y_i$ defining $X_\\xi$ are never zero, i.e. that for each $[x_0:x_1:x_2]\\in \\p^2$ we cannot have $x_0^2+ 2\\xi x_1x_2=x_1^2+ 2\\xi x_0x_2=x_2^2+ 2\\xi x_0x_1=0.$ This follows from Lemma \\[lem:ThreeConics\\] and from the hypothesis $\\xi^3\\not=-\\frac{1}{8}$.\n\nThe equation of $X_\\xi$ is given by $$( x_0 \\, x_1 \\, x_2 ) \\cdot M \\cdot \\begin{pmatrix} x_0 \\\\ x_1 \\\\ x_2\\end{pmatrix}=0\n\\text{ with }\nM=\\begin{pmatrix}\ny_0& \\xi y_2& \\xi y_1\\\\\n\\xi y_2& y_1& \\xi y_0\\\\\n \\xi y_1& \\xi y_0& y_2\n\\end{pmatrix} \\in \\Mat_{3\\times 3}(\\C[y_0,y_1,y_2]).$$ The polynomial $\\det(M)$ is equal to $$\\det(M)=\\lambda (y_0^3+y_1^3+y_2^3)+\\mu y_0y_1y_2,\\text{ with }\\lambda=-\\xi^2\\text{ and }\\mu=2\\xi^3+1.$$ In particular, the fibres of the second projection $X_\\xi/ \\p^2$ are all conics (the coefficients of $x_i^2$ is $y_i$ so not all coefficients can be zero) and a general one is irreducible. ", "As the threefold $X_\\xi$ is smooth, irreducible and satisfies $\\rho(X_\\xi)=2$, the morphism $X_\\xi/ \\p^2$ is a standard conic bundle. ", "Its discriminant is given by the zero locus of $\\det(M)$, which is a polynomial of degree $3$ which has the classical Hesse Form. ", "The discriminant corresponds to a smooth cubic if $\\lambda(27\\lambda^3+\\mu^3)\\not=0$, and to the union of three lines in general position otherwise. ", "To prove this classical fact, we compute the partial derivatives of $\\det(M)$, which are $(3\\lambda y_0^2+\\mu y_1y_2,3\\lambda y_1^2+\\mu y_0y_2,3\\lambda y_2^2+\\mu y_0y_1).$ By Lemma \\[lem:ThreeConics\\], this has no zeroes in $\\p^2$ if $\\lambda(27\\lambda^3+\\mu^3)\\not=0$ and has three zeroes otherwise. ", "It remains to observe that $27\\lambda^3+\\mu^3=(8\\xi^3+1)(\\xi^3-1)^2$.\n\nLet $\\k$ be a subfield of $\\C$ and $\\xi\\in\\k$. Then the curve $\\Delta_\\xi$ of Lemma \\[lem:Xxi\\] is defined over $\\k$ and has a $\\k$-rational point, namely the inflexion point $[0:1:-1]$. When $\\k=\\C$, one can prove that all elliptic curves are obtained in this way; for smaller fields this does not seem to be true. ", "We will however show that there are enough such curves.", "\n\nWe thank P. Habegger for helpful discussions concerning the next lemma.", "\n\n\\[lem:Isogenies\\] Let $\\k\\subseteq \\C$ be a subfield.", "\n\n1. ", " \\[jinvariant\\] For each $\\xi\\in \\k$, with $\\xi^3\\notin \\{0,-\\frac{1}{8},1\\}$, we denote $($as in Lemma $\\ref{lem:Xxi})$ by $\\Delta_\\xi$ the smooth cubic curve defined over $\\k$ given by $$-\\xi^2(y_0^3+y_1^3+y_2^3)+(2\\xi^3+1)y_0y_1y_2=0.$$ The $j$-invariant of $\\Delta_\\xi$ is equal to $$\\left(\\dfrac{16\\xi^{12}+464\\xi^9+240\\xi^6+8\\xi^3+1}{\\xi^2(8\\xi^9-15\\xi^6+6\\xi^3+1)}\\right)^3.$$\n\n2. ", " \\[isogenies\\] There is a subset $J\\subseteq \\k$ having the same cardinality as $\\k$ such that for all $\\xi,\\xi'\\in J$, the following are equivalent:\n\n 1. ", " There exist surjective morphisms $\\Delta_\\xi\\tto \\Delta_{\\xi'}$ and $\\Delta_{\\xi'}\\tto \\Delta_{\\xi}$ defined over $\\C$;\n\n 2. ", " $\\xi=\\xi'$.\n\n\\[jinvariant\\]. ", "By Lemma \\[lem:Xxi\\], $\\Delta_\\xi$ is a smooth cubic curve if $\\xi^3\\notin \\{0,-\\frac{1}{8},1\\}$. We choose the inflexion point $[0:1:-1]\\in \\Delta_\\xi$ to be the origin, make a coordinate change so that the inflexion line is the line at infinity, and thusly obtain a Weierstrass form. ", "Then we compute the $j$-invariant as in [@silverman .1 page 42]; this is tedious but straightforward. ", "This can also be done using the formulas from [@ArtebaniDolgachev page 240].", "\n\n\\[isogenies\\]. ", "Let $\\xi,\\xi'\\in \\k$ be such that $\\xi^3,(\\xi')^3\\notin \\{0,-\\frac{1}{8},1\\}$. We see the curves $\\Delta_\\xi$ and $\\Delta_{\\xi'}$ as elliptic curves defined over $\\k$ with origin $O=[0:1:-1]$. Suppose that there is a surjective morphism $\\phi\\colon \\Delta_\\xi\\tto \\Delta_{\\xi'}$ defined over $\\C$. It sends the origin of $\\Delta_\\xi$ onto a $\\C$-rational point of $\\Delta_{\\xi'}$. Applying a translation at the target, we can assume that $\\phi(O)=O$, which means that $\\phi$ is an isogeny, and that $\\Delta_{\\xi}$ and $\\Delta_{\\xi'}$ are isogenous over $\\C$ (see [@silverman Definition, §III.4 page 66]).", "\n\nWe now choose a sequence $p_1,p_2,\\ldots$ of increasing prime numbers such that for each $i\\ge 2$, the prime number $p_i$ does not appear in the denominator of the $j$-invariant of $\\Delta_{p_{i'}}$ for each $i'<i$. For each $i\\ge 1$, the $j$-invariant of $\\Delta_{p_i}$ is an element of $\\Q$ having a denominator divisible by $p_i$ (follows from \\[jinvariant\\]), so $\\Delta_{p_i}$ does not have potential good reduction modulo $p_i$ but this does not hold for $\\Delta_{p_{i'}}$ for $i'>i$, which then has potential good reduction modulo $p_{i}$ [@silverman Proposition 5.5, §VII.5, page 197]. ", "This implies that there is no isogeny $\\Delta_{p_i}\\to \\Delta_{p_{i'}}$ defined over any number field $K$ and where one curve has good reduction and the other has bad reduction [@silverman Corollary 7.2, §VII.7, page 202], and thus no isogeny defined over $\\C$ [@MasserWustholz Lemma 6.1]. ", "If $\\k$ is countable, this achieves the proof of \\[isogenies\\].", "\n\nIt remains to consider the case where $\\k$ is an uncountable subfield of $\\C$. The set of $j$-invariants of curves $\\Delta_\\xi$, where $\\xi\\in \\k$ is such that $\\xi^3\\notin \\{0,-\\frac{1}{8},1\\}$, is then uncountable too.", "\n\nWe denote by $\\Omega\\subseteq \\C^2$ the set consisting of pairs $(j_1,j_2)\\in \\widebar{\\Q}^2$ such that the curves of $j$-invariants $j_1$ and $j_2$ are isogenous. ", "The set $\\Omega$ is a countable union of algebraic curves defined over $\\Q$, given by the zero set of the so-called modular transformation polynomials (see [@lang 5§3] and in particular [@lang Theorem 5, Chapter 5§3, page 59]). ", "Moreover, these curves are irreducible and invariant under the exchanges of variables $(x,y)\\mapsto (y,x)$ [@lang Theorem 3, Chapter 5§3, page 55], so are not vertical or horizontal lines in $\\C^2$.\n\nWe write $S=\\{\\xi\\in \\k\\mid \\xi^3\\notin \\{0,-\\frac{1}{8},1\\}\\}$. Then, by the previous paragraph, for each element $\\xi\\in S$ the curve $\\Delta_\\xi$ is isogeneous (over $\\C$) to only countably many isomorphism classes of $\\Delta_{\\xi'}$ with $\\xi'\\in\\k$. Putting an equivalence relation on $S$ saying that two elements are equivalent if the curves are isogeneous over $\\C$ (see [@silverman III.6, Theorem 6.1(a)]), we obtain that each equivalence class is countable, so the set of equivalence classes has the cardinality of $S$, or equivalently of $\\k$. This achieves the proof.", "\n\n\\[pro:EnoughCB\\] Let $\\k$ be a subfield of $\\C$. For each $n\\ge 3$, there is a set $J$ having the cardinality of $\\k$ indexing decomposable conic bundles $X_i/B_i$ defined over $\\k$, where $X_i,B_i$ are smooth varieties rational over $\\k$, and such that two conic bundles $X_i/B_i$ and $X_j/B_j$ are equivalent $($over $\\C)$ if and only if $i=j$.\n\nWe choose the set $J\\subseteq \\k$ of Lemma \\[lem:Isogenies\\]\\[isogenies\\], and consider, for each $\\xi\\in J$, the hypersurface $X_\\xi\\subset \\p^2\\times \\p^2$ of Lemma \\[lem:Xxi\\], which is given by $$X_\\xi=\\Bigl\\{([x_0:x_1:x_2],[y_0:y_1:y_2])\\in \\p^2 \\times \\p^2 \\Bigm| \\sum_{i=0}^2 (x_i^2+ 2\\xi \\frac{x_0x_1x_2}{x_i})y_i=0\\Bigr\\}$$ By Lemma \\[lem:Xxi\\], the second projection gives a standard conic bundle $X_\\xi\\to \\p^2$ whose discriminant curve $\\Delta_\\xi\\subset \\p^2$ is given by $-\\xi^2(y_0^3+y_1^3+y_2^3)+(2\\xi^3+1)y_0y_1y_2$. Note that $(X_\\xi \\times \\p^{n-3})/(\\p^2\\times \\p^{n-3})$ (or simply $X_\\xi/\\p^2$ if $n=3$) is a decomposable conic bundle defined over $\\k$, as it is embedded in the trivial $\\p^2$-bundle $(\\p^2\\times \\p^2\\times \\p^{n-3})/(\\p^2\\times \\p^{n-3})$ by construction. ", "Moreover, $X_\\xi \\times \\p^{n-3}$ is birational to $\\p^n$ over $\\k$ (Lemma \\[lem:Xxi\\]). ", "By Proposition \\[pro:Isogenies\\], two conic bundles $(X_\\xi \\times \\p^{n-3})/(\\p^2\\times \\p^{n-3})$ and $(X_{\\xi'} \\times \\p^{n-3})/(\\p^2\\times \\p^{n-3})$ are equivalent only if there exist surjective morphisms $\\Delta_\\xi\\tto \\Delta_{\\xi'}$ and $\\Delta_{\\xi'}\\tto \\Delta_{\\xi}$. This is only possible if $\\xi=\\xi'$, by Lemma \\[lem:Isogenies\\]\\[isogenies\\].", "\n\nProofs of Theorems \\[TheoremManyCB\\] and \\[TheoremTame\\]\n--------------------------------------------------------\n\nBy Theorem \\[TheoremBirMori\\], we have a group homomorphism and a groupoid homomorphism $$\\begin{tikzcd}[link]\n\\Bir(\\p^n) \\ar[rr]&&\\bigast_{C\\in \\CB(\\p^n)} \\left(\\bigoplus_{\\MC(C)}\\Z/2\\right) \\\\\n\\BirMori(\\p^n) \\ar[u,phantom,\"\\vertsubseteq\"]\\ar[urr]&\n\\end{tikzcd}$$ For each subfield $\\k\\subseteq \\C$, we can embed $\\Bir_\\k(\\p^n)$ into $\\Bir_\\C(\\p^n)$ and look at the image in $\\bigast_{C\\in \\CB(\\p^n)} \\left(\\bigoplus_{\\MC(C)}\\Z/2\\right)$. We consider the set of decomposable conic bundles $X_i/B_i$ defined over $\\k$ indexed by $J$ of Proposition \\[pro:EnoughCB\\], which give pairwise distinct elements of $C_i\\in \\CB(\\p^n)$, and associate to these birational maps $\\psi_i\\colon X_i\\rat \\p^n$ defined over $\\k$. For each $i\\in J$, there is an involution $\\iota_i\\in\\psi_i\\Bir_\\k(X_i/B_i)\\psi_i^{-1}\\subseteq \\Bir_\\k(\\p^n)$ whose image in $\\bigoplus_{\\MC(C_i)}\\Z/2$ is not trivial by Proposition \\[pro:largeimageAnyCB\\]. ", "One can thus take a projection $\\bigoplus_{\\MC(C_i)}\\Z/2\\to \\Z/2$ such that the image of $\\iota_i$ is non-trivial. ", "We obtain a surjective group homomorphism from $\\Bir_\\k(\\p^n)$ to $\\smash{\\bigast_{i\\in J} \\Z/2}$ where $J$ has the cardinality of $\\k$ and such that each involution $\\iota_i\\in \\Bir_\\k(\\p^n)$ is sent onto the generator indexed by $i$. There is thus a section of this surjective group homomorphism.", "\n\n\\[rem:product\\_and\\_sum\\] As Proposition \\[pro:EnoughCB\\] gives an infinite image, the above proof naturally gives a surjective homomorphism to the group $\\bigast_{J} (\\bigoplus_\\Z \\Z/2)$, but since there is an abstract surjective homomorphism from $\\bigast_{J} \\Z/2$ to this group, we chose not to mention the direct sum in the statement of the theorem.", "\n\nMoreover, with the alternative form the existence of a section would be far less clear. ", "Indeed, $(\\Z/2)^3$ does not embed in $\\Bir(X/B)$ and $(\\Z/2)^7$ does not embed in $\\Bir(X)$, for $X$ rationally connected of dimension $3$ [@Pro2011; @Pro2014], so it seems probable that $\\bigoplus_\\Z \\Z/2$ does not embed in $\\Bir(X)$ for any variety $X$.\n\nWe consider a subfield $\\k$ of $\\C$, an integer $n\\ge 3$, and a subset $S\\subset \\Bir_\\k(\\p^n)$ of cardinality smaller than the one of $\\k$. We want to construct a surjective homomorphism $\\Bir_\\k(\\p^n) \\tto \\Z/2$ such that the group $G$ generated by $\\Aut_\\k(\\p^n)$, by all Jonquières elements and by $S$ is contained in the kernel. ", "We use the group homomorphism $$\\tau\\colon \\Bir_\\k(\\p^n) \\to \\bigast_J \\Z/2$$ given by Theorem \\[TheoremManyCB\\]. ", "Each $j \\in J$ corresponds to a conic bundle $X_j/B_j$. The group $\\Aut_\\k(\\p^n)$ is in the kernel of $\\tau$. The group of Jonquières elements is conjugated to the subgroup $J\\subset \\Bir(\\p^1 \\times \\p^{n-1})$ consisting of elements sending a general fibre of $\\p^1\\times \\p^{n-1}/\\p^{n-1}$ onto another one. ", "The action on the base yields an exact sequence $$1\\to \\Bir(\\p^1 \\times \\p^{n-1}/\\p^{n-1})\\to J\\to \\Bir(\\p^{n-1})\\to 1.$$ This gives $J=\\Bir(\\p^1 \\times \\p^{n-1}/\\p^{n-1}) \\rtimes J'$, where $J'\\subset J$ is the group isomorphic to $\\Bir(\\p^{n-1})$ that acts on $\\p^1\\times \\p^{n-1}$ with trivial action on the first factor. ", "We can assume that $\\p^1 \\times \\p^{n-1}/\\p^{n-1} = X_{j_0}/B_{j_0}$ for some $j_0 \\in J$. The image of $\\Bir(\\p^1 \\times \\p^{n-1}/\\p^{n-1})$ by $\\tau$ is contained in the group $\\Z/2$ indexed by $j_0$. Now observe that $J'\\subset \\Ker\\tau$. Indeed, we first decompose an element of $J'\\simeq \\Bir(\\p^{n-1})$ as a product of Sarkisov links between Mori fibre spaces $Y_i\\to S_i$, where $Y_i$ has dimension $n-1$, and observe that taking the product with $\\p^1$ gives Sarkisov links between the Mori fibre spaces $Y_i\\times \\p^1 \\to S_i\\times \\p^1$ of dimension $n$. Each of the Sarkisov links of type $\\II$ arising in such decomposition has covering gonality $1$, as $\\covgon(\\Gamma\\times \\p^1)=1$ for each variety $\\Gamma$.\n\nWe consider the group homomorphism $\\hat\\tau\\colon \\Bir_\\k(\\p^n)\\to \\bigast_{J \\setminus \\{j_0\\}}\\Z/2$ obtained by composing $\\tau$ with the projection $\\bigast_{J}\\Z/2\\to \\bigast_{J \\setminus \\{j_0\\}}\\Z/2$ obtained by forgetting the factor indexed by $j_0$.\n\nThe image by $\\hat\\tau$ of all Jonquières elements is trivial, hence the group $\\hat\\tau(G)$ has at most the cardinality of $S$, which by assumption is strictly smaller than the cardinality of $J$. We construct the expected morphism by projecting from $\\hat\\tau(\\Bir_\\k(\\p^n))$ onto a factor $\\Z/2$ which is not in the image of $G$.\n\nComplements {#sec:Complements}\n===========\n\nQuotients and SQ-universality {#sec:SQ}\n-----------------------------\n\nA direct consequence of Theorem \\[TheoremManyCB\\] is that we have a lot of quotients of $\\Bir_\\k(\\p^n)$ for $n\\ge 3$.\n\nFirstly, we can have quite small quotients (which is not the case for $\\Bir_\\C(\\p^2)$ which has no non-trivial countable quotient, as mentioned before):\n\nFor each $n\\ge 3$, each subfield $\\k\\subseteq \\C$, and each integer $m\\ge 1$, there are $($abstract$)$ surjective group homomorphisms from $\\Bir_\\k(\\p^n)$ to the dihedral group $D_{2m}$ of order $2m$ and the symmetric group $\\Sym_m$. In particular, there is a normal subgroup of $\\Bir_\\k(\\p^n)$ of index $r$ for each even integer $r> 1$.\n\nFollows from Theorem \\[TheoremManyCB\\] and the fact that $D_{2m}$ and $\\Sym_m$ are generated by involutions.", "\n\nSecondly, we get much larger quotients:\n\nFor any $n\\ge 3$, any subfield $\\k\\subseteq \\C$ and any integer $m\\ge 1$, there are $($abstract$)$ surjective group homomorphisms $$\\begin{aligned}\n\\Bir_\\k(\\p^n)\\tto \\SL_m(\\k),&&\\Bir_\\k(\\p^n)\\tto \\Bir_{\\overline{\\Q}}(\\p^2).\\end{aligned}$$\n\nWe observe that $\\SL_m(\\k)$ has the cardinality of $\\k$ and that $\\Bir_{\\overline{\\Q}}(\\p^2)$ is countable. ", "Hence, both groups have at most the cardinality of $\\k$. Both groups are generated by involutions: for $\\Bir_{\\overline{\\Q}}(\\p^2)$ this is by the Noether-Castelnuovo Theorem which says that $\\Bir_{\\overline{\\Q}}(\\p^2)$ is generated by the standard quadric involution and by $\\Aut_{\\overline{\\Q}}(\\p^3)\\simeq \\PGL_3(\\overline{\\Q})=\\PSL_3(\\overline{\\Q})$, and thus is generated by involutions. ", "Hence, the two groups are quotients of $\\bigast_{J} \\Z/2$. The result then follows from Theorem \\[TheoremManyCB\\].", "\n\nSimilarly, over $\\C$ we get:\n\nFor any $n \\ge 3$, there exists a surjective group homomorphism $$\\Bir_\\C(\\p^n)\\tto \\Bir_\\C(\\p^2).$$\n\nRecall that a group $G$ is *SQ-universal* if any countable group embeds in a quotient of $G$. The free group $\\Z * \\Z$ was an early example of SQ-universal group. ", "More generally any nontrivial free product $G_1 * G_2$ distinct from $\\Z/2 * \\Z/2$ is SQ-universal, see [@Schupp Theorem 3]. ", "From a modern point of view, this also follows from [@MinasyanOsin], by looking at the action of any loxodromic isometry on the associated Bass-Serre tree. ", "In particular, taking $G_1 = \\Z/2 * \\Z/2$ and $G_2 = \\Z/2$, we get that $\\Z/2 * \\Z/2 * \\Z/2$ is SQ-universal.", "\n\nFor any field $\\k\\subseteq \\C$ and any $n \\ge 3$, the Cremona group $\\Bir_\\k(\\p^n)$ admits a surjective morphism to the SQ-universal group $\\Z/2 * \\Z/2 * \\Z/2$. In particular, $\\Bir_\\k(\\p^n)$ also is SQ-universal.", "\n\nFollows from Theorem \\[TheoremManyCB\\] and from the fact that $\\Z/2 * \\Z/2 * \\Z/2$ is SQ-universal.", "\n\nHopfian property {#sec:Hopfian}\n----------------\n\nRecall that a group $G$ is *hopfian* is every surjective group homomorphism $G\\tto G$ is an isomorphism. ", "It was proven in [@Deserti] that the group $\\Bir_\\C(\\p^2)$ is hopfian. ", "An open question, asked by I. Dolgachev (see [@Des17]), is whether the Cremona group $\\Bir_\\C(\\p^n)$ is generated by involutions for each $n$, the answer being yes in dimension $2$ and open in dimension $\\ge 3$. Theorem \\[TheoremManyCB\\] relates these two notions and shows that we cannot generalise both results at the same time (being hopfian and generated by involutions) to higher dimension.", "\n\n\\[Cor:Hopfian\\] For each $n\\ge 3$ and each subfield $\\k\\subseteq \\C$, the group $\\Bir_\\k(\\p^n)$ is not hopfian if it is generated by involutions.", "\n\nFollows from Theorem \\[TheoremManyCB\\], as the group homomorphisms provided by Theorem \\[TheoremManyCB\\] is not injective, and because $\\Bir_\\k(\\p^n)$ has the same cardinality as $\\k$ (the set of all polynomials of degree $n$ with coefficients in $\\k$ has the same cardinality as $\\k$).", "\n\nMore general fields {#sec:field}\n-------------------\n\nEvery field isomorphism $\\k\\iso\\k'$ naturally induces an isomorphism $\\Bir_\\k(\\p^n)\\iso \\Bir_{\\k'}(\\p^n)$. More generally, it associates to each variety and each rational map defined over $\\k$, a variety and a rational map defined over $\\k'$. It then induces an isomorphism between the group of birational maps defined over $\\k$ and $\\k'$ of the varieties obtained. ", "This implies that the five Theorems \\[TheoremBirPnnotsimple\\]-\\[TheoremManyCB\\] also hold for each ground field which is abstractly isomorphic to a subfield of $\\C$. This includes any field of rational functions of any algebraic variety defined over a subfield of $\\C$ as these fields have characteristic zero and cardinality smaller or equal than the one of $ \\C$.\n\nAmalgamated product structure\n-----------------------------\n\nWe work over the field $\\C$. In the next result, an element of $\\CB(X)$ is said to be decomposable if it is the class of a decomposable conic bundle (in the sense of Definition \\[def:DecomposableCB\\]).", "\n\n\\[thm:AmalgamatedProduct\\] For each integer $n\\ge 3$, and let $X/B$ be a conic bundle, where $X$ is a terminal variety of dimension $n$. We denote by $\\rho$ the group homomorphism $$\\rho\\colon \\Bir(X) \\to \\bigast_{C\\in \\CB(X)} \\left(\\bigoplus_{\\MC(C)}\\Z/2\\right)$$ given by Theorem \\[TheoremBirMori\\]. ", "For each $C\\in \\CB(X)$ we fix a choice of representative $X_C/B_C$, and we denote $G_C = \\rho^{-1} (\\rho (\\Bir(X_C/B_C)))\\subseteq \\Bir(X)$. Then, the following hold:\n\n1. ", " \\[IntersectA\\] For all $C \\neq C'$ in $\\CB(X)$, the group $A = G_C \\cap G_{C'}$ contains $\\ker \\rho$ and does not depend on the choice of $C,C'$;\n\n2. ", " \\[BirXFreeproduct\\] The group $\\Bir(X)$ is the free product of the groups $G_C$, $C\\in \\CB(X)$, amalgamated over their common intersection $A$: $$\\Bir(X) = \\bigast_{A} G_C.$$\n\n3. ", " \\[NonTrivial\\] For each decomposable $C\\in \\CB(X)$ we have $A\\subsetneq G_C$. Moreover, the free product of \\[BirXFreeproduct\\] is non-trivial $($i.e. $A\\subsetneq G_C\\subsetneq \\Bir(X)$ for each $C)$ as soon as $\\CB(X)$ contains two distinct decomposable elements. ", "This is for instance the case when $X$ is rational, as $\\CB(X)$ then contains uncountably many decomposable elements.", "\n\n\\[IntersectA\\]. ", "For each $C\\in \\CB(X)$, we denote by $H_C=\\left(\\bigoplus_{\\MC(C)}\\Z/2\\right)$ the factor indexed by $C$ in the free product $\\bigast_{C\\in \\CB(X)} \\left(\\bigoplus_{\\MC(C)}\\Z/2\\right)=\\bigast_{C\\in \\CB(X)} H_C$. By definition of the group homomorphism, for each $C\\in \\CB(X)$ we have $\\rho(\\Bir(X_C/B_C)) \\subseteq H_C$. As $H_C$ is a $\\F_2$-vector space with basis $\\MC(C)$ and $\\rho(\\Bir(X_C/B_C))$ is a linear subspace, there exists a projection $H_C\\to \\rho(\\Bir(X_C/B_C))$. We then denote by $$\\rho'\\colon \\Bir(X) \\to \\bigast_{C\\in \\CB(X)} \\rho(\\Bir(X_C/B_C))$$ the group homomorphism induced for each $C$ by the projection $H_C\\to\\rho(\\Bir(X_C/B_C))$. By definition of the free product, we obtain $H_C\\cap H_{C'}=\\id$ for all $C\\not=C'$. This implies that $G_C \\cap G_{C'} = \\ker \\rho' \\supseteq \\ker \\rho$.\n\n\\[BirXFreeproduct\\]. ", "We first observe that by construction the groups $G_C$ generate the group $\\Bir(X)$. The fact that $\\Bir(X) = \\bigast_{A} G_C$ corresponds to saying that all relations in $\\Bir(X)$ lie in the groups $G_C$. This follows from the group homomorphism $\\rho$ to a free product, where no relation between the groups $H_C$ exists.", "\n\n\\[NonTrivial\\]. ", "The fact that $A\\subsetneq G_C$ for each decomposable $C$ follows from Proposition \\[pro:largeimageAnyCB\\]. ", "Hence, the free product of \\[BirXFreeproduct\\] is non-trivial if there are least two $C$ corresponding to decomposable conic bundles. ", "If $X$ is rational, then we moreover have uncountably many such elements by Proposition \\[pro:EnoughCB\\].", "\n\nIn Theorem \\[thm:AmalgamatedProduct\\], one could be tempted to say that $A=\\ker\\rho$, but this is not clear. ", "Indeed, it could be that some elements of $\\bigoplus_{\\MC(C)}\\Z/2$ are in the image of $\\Bir(X)$ but not in the image of $\\Bir(X/B)$.\n\nCubic varieties {#Sec:CubicVar}\n---------------\n\nHere again we work over $\\C$. We recall the following result, which allows to apply Theorem \\[TheoremBirXBnotsimple\\] to any smooth cubic hypersurface of dimension $\\ge 3$:\n\n\\[Lem:SmoothCubicHypersurfaceLine\\] Let $n\\ge 4$ and let $\\ell\\subset X\\subset \\p^n$ be a line on a smooth cubic hypersurface. ", "We denote by $\\hat{X}$ and $P$ the respective blow-ups of $X$ and $\\p^n$ along $\\ell$. Then, the projection away from $\\ell$ gives rise to a decomposable conic bundle and a decomposable $\\p^2$-bundle $$\\hat{X}\\subset P=\\p(\\mathcal{O}_{\\p^{n-2}}\\oplus \\mathcal{O}_{\\p^{n-2}}\\oplus \\mathcal{O}_{\\p^2}(1))\\to \\p^{n-2}.$$ Moreover, the discriminant of the conic bundle is a hypersurface of degree $5$.\n\nWe take coordinates $[y_0:y_1:\\cdots:y_{n-2}:u:v]$ on $\\p^n$ and assume that $\\ell\\subset \\p^n$ is the line given by $y_0=y_1=\\cdots=y_{n-2}=0$. The equation of $X$ is then given by $$Au^2+2Buv+Cv^2+2Du+2Ev+F=0$$ where $A,B,C,D,E,F\\in \\C[y_0,\\ldots,y_{n-2}]$ are homogeneous polynomials of degree $1,1,1,2,2,3$ respectively.", "\n\nAs in the proof of Proposition \\[pro:largeimageAnyCB\\], we view $P=\\p(\\mathcal{O}_{\\p^{n-2}}\\oplus \\mathcal{O}_{\\p^{n-2}}\\oplus \\mathcal{O}_{\\p^{n-2}}(1))$ as the quotient of $(\\A^{2}\\setminus \\{0\\})\\times (\\A^{n-1}\\setminus \\{0\\})$ by $(\\mathbb{G}_m)^2$ via $$\\left((\\lambda,\\mu),(x_0,x_1,x_2,y_0,y_1,\\cdots,y_{n-2})\\right)\\mapsto (\\lambda x_0,\\lambda x_1,\\lambda\\mu^{-1} x_2,\\mu y_0,\\cdots,\\mu y_{n-2})$$ and denote by $[x_0\\colon x_1 \\colon x_2 \\semicolon y_0 \\colon \\cdots \\colon y_{n-2}]\\in P$ the class of $(x_0,x_1,x_2,y_0,\\cdots,y_{n-2})$. The birational morphism $$\\begin{tikzcd}[map]\nP & \\to & \\p^n\\\\\n{[x_0\\colon x_1 \\colon x_2 \\semicolon y_0 \\colon y_1 \\colon y_2\\colon \\cdots \\colon y_{n-2}]} & \\mapsto & {[x_2y_0 \\colon \\cdots \\colon x_2y_{n-2}:x_0:x_1]}\n\\end{tikzcd}$$ is the blow-up of $\\ell$, so $\\hat{X}$ is given by $$Ax_0^2+2Bx_0x_1+Cx_1^2+2Dx_2x_0+2Ex_2x_1+Fx_2^2=0,$$ which is then a conic bundle over $\\p^2$. The discriminant of the curve gives a hypersurface $\\Delta\\subset \\p^2$ of degree $5$, given by the determinant of $\\left(\\begin{smallmatrix} A & B & D \\\\\n B & C & E \\\\ D & E & F\\end{smallmatrix}\\right)$.\n\n\\[cor:SmoothCubicBirXZ2\\] For each $n\\ge 4$ and each smooth cubic hypersurface $X\\subset \\p^n$, there exists a surjective group homomorphism $\\Bir(X)\\tto \\bigoplus_\\Z \\Z/2$\n\nFollows from the application of Theorem \\[TheoremBirXBnotsimple\\] to the conic bundle associated to blow-up a line of $X$ (Lemma \\[Lem:SmoothCubicHypersurfaceLine\\]).", "\n\nEvery smooth cubic threefold $X\\subset \\p^4$ is not rational, and moreover two such cubics are birational if and only if they are projectively equivalent, i.e. equal up to an element of $\\Aut(\\p^4)=\\PGL_5(\\C)$ [@ClemensGriffiths]. ", "We moreover get the following:\n\n\\[Prop:Cubicfreeproduct\\] Let $X\\subset \\p^4$ be a general smooth cubic hypersurface. ", "We have a surjective group homomorphism $\\Bir(X)\\tto \\bigast_J \\Z/2$, where $J$ has the cardinality of $\\C$.\n\nThe map of Lemma \\[Lem:SmoothCubicHypersurfaceLine\\] associates to each smooth cubic threefold $X$ and each line $\\ell\\subset X$ a quintic curve $\\Delta\\subset \\p^2$ and also a theta-characteristic; this induces a birational map between the pairs $(\\ell,X)$ of lines on smooth cubic threefolds, up to $\\PGL_5(\\C)$, and the pairs $(\\theta,\\Delta)$, where $\\Delta\\subset \\p^2$ is a smooth quintic and $\\theta$ is a theta-characteristic, again up to $\\PGL_3(\\C)$ [@CasRob Theorem 4.1 and Proposition 4.2].", "\n\nIn particular, taking a general smooth cubic hypersurface $X\\subset \\p^4$ and varying the lines $\\ell\\subset X$ (which form a $2$-dimensional family), we obtain a family $J$ of dimension $2$ of smooth quintics $\\Delta\\subset \\p^2$, not pairwise equivalent modulo $\\PGL_3(\\C)$. This yields conic bundles that are not pairwise equivalent, parametrised by a complex algebraic variety of dimension $2$. Applying the group homomorphism of Theorem \\[TheoremBirMori\\] and projecting on the corresponding factors provides a surjective group homomorphism $\\Bir(X)\\tto \\bigast_J \\Z/2$, similarly as in the proof of Theorem \\[TheoremManyCB\\].", "\n\nFibrations graph {#sec:graph}\n----------------\n\nWe explain how to get a natural graph structure from the set of rank $r$ fibrations, similarly as in [@LZ].", "\n\nLet $Z$ be a variety birational to a Mori fibre space. ", "We construct a sequence of nested graphs $\\Gl_n$, $n \\ge 1$, as follows. ", "The set of vertices of $\\Gl_n$ are rank $r$ fibrations $X/B$, for any $r \\le n$, with a choice of a birational map $\\phi\\colon Z \\rat X$, and modulo $Z$-equivalence (Definition \\[def:Tequivalent\\]). ", "We denote $(X/B, \\phi)$ such an equivalence class. ", "We put an oriented edge from $(X/B, \\phi)$ to $(X'/B', \\phi')$ if $\\rho(X'/B') = \\rho(X/B) - 1$ and the birational maps from $Z$ induce a factorisation of $X/B$ through $X'/B'$, that is, if there is a morphism $B' \\to B$ and a birational contraction $X \\rat X'$ such that the following diagram commutes $$\\begin{tikzcd}[link]\n& Z \\ar[dr, dashed,\"\\phi'\"] \\ar[dl, dashed, \"\\phi\",swap] \\\\\nX \\ar[rr,dashed] \\ar[dd] && X' \\ar[dd] \\\\ \\\\\nB && B' \\ar[ll]\n\\end{tikzcd}$$ We call the graph $\\Gl := \\bigcup_n \\Gl_n$ the *fibrations graph* associated with $Z$. The group $\\Bir(Z)$ naturally acts on each graph $\\Gl_n$, and so also on $\\Gl$, by precomposition : $$g \\cdot (X/B, \\phi) := (X/B, \\phi \\circ g^{-1}).$$\n\nThe fact that Sarkisov links generate $\\BirMori(Z)$ is equivalent to the fact that $\\Gl_2$ is a connected graph. ", "Lemma \\[lem:2 rank 2\\] implies that $\\Gl_3$ is the $1$-skeleton of a square complex, where each square has one vertex of rank 3, one vertex of rank 1 and two vertices of rank 2. ", "The fact that elementary relations generate all relations in $\\BirMori(Z)$ is equivalent to the fact that this square complex is simply connected.", "\n\nIt is not clear to us if for $n \\ge 4$ the graph $\\Gl_n$ is still the $1$-skeleton of a cube complex.", "\n\n10000\n\n[^1]: “However, other group-theoretic questions related to the Cremona group of the plane (and, even more so, of $\\p^n$, $n>2$) remain unsolved; for example, the important question of whether the Cremona group contains any normal subgroup (a question which seems likely to be answered negatively).”", "\n\n[^2]: “This theorem can not be easily extended to $\\p^n$ where $n>2$; therefore, the main question of finding the most simple generating transformations of the entire Cremona group of $\\p^n$ for $n>2$ remains open.”", "\n" ]

{ "pile_set_name": "ArXiv" }

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[ "QOS Networks is seeking a Frontend Software Engineer to join its analytics and platform services team.", "\n\nWe are leading the disruption that is occurring in branch management and application monitoring infrastructure. ", "The increased adoption of cloud & SaaS applications coupled with the dramatic growth of rich media on corporate networks is causing enterprises to rethink how they architect their WANs. ", "Hybrid WAN networks, which combine traditional MPLS WAN circuits with direct-to-Internet circuits, have emerged in an attempt to achieve the right balance of cost & performance when accessing both on-premises & SaaS-based applications. ", "SD-WAN is a fundamentally new approach to WAN architectures that provides a simpler, more agile way to manage both hybrid and traditional WAN, alike.", "\n\nYou will be responsible for building the next generation of tools to manage, visualize, automate and control these cutting-edge network technologies.", "\n\nResponsibilities\n\n\n\nImplement the frontend experience for our customer-facing analytics, management and visibility platforms and tools\n\n\n\nWork closely with UX designers and backend developers to architect and implement modern web applications\n\n\n\nDesign and develop UI patterns and practices used throughout QOS Networks projects\n\n\n\nRequirements\n\n\n\nBachelor’s degree or equivalent software work experience\n\n\n\n3+ years developing modern web applications using one or more major frameworks, such as React or Angular\n\n\n\nProficiency in styling, layout and asset management for responsive designs\n\n\n\nHigh level of hands on implementation of projects using JavaScript, with experience in ES6 and transpilation\n\n\n\nRecommended Qualifications" ]

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[ "This blog is dedicated to climate change, biodiversity and general environmental sustainability issues that I find relevant or interesting. ", "It may develop into a good place to visit regularly if you are involved in biodiversity and climate change issues in the Western Cape.", "\n(Best viewed using Mozilla Firefox - http://www.mozilla.com)\n\n19 June, 2006\n\nCFR Environmental Vacancies Blog ...\n\nI thought it may be useful to post all these on a dedicated blog rather than this Biodiversity and Environmental Sustainability blog, so I have, on a trial basis, created a CFR Environmental vacancies Blog at http://cfkjobs.blogspot.com/\n\nI have included a link to the CFR Environmental Vacancies Blog under 'Links' 'My Sites' on the sidebar." ]

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[ "Ohio's 18th senatorial district\n\nOhio's 18th senatorial district has been established in northeastern Ohio, primarily along the shore of Lake Erie. ", " It currently consists of Portage County and portions of Lake and Geauga counties. ", " It encompasses Ohio House districts 61, 75 and 76. ", " It has a Cook PVI of R+1. ", " Its current Ohio Senator is Republican John Eklund. ", " He resides in Chardon, a city located in Geauga County.", "\n\nList of senators\n\nReferences\n\nExternal links\nOhio's 18th district senator at the 133rd Ohio General Assembly official website\n\nCategory:Ohio State Senate districts" ]

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[ "Q:\n\nMake someone esle pay for Gas: Transaction Address Signer and Transaction Address Injector are not the same\n\nIs there a way to sign a transaction by a user, and apply it by an other so that the second account pays the gas?", "\nIn two words: The transaction signer is Bob, the transaction injector in the Blockchain is Alice for example.", "\nJust like in Ethereum with web3.js, we can use \"encodeABI\" method to encode the transaction and sign it by a user and apply (inject it in the blockchain) it by an other.", "\n\nA:\n\nYou can do it with a smart contract. ", "Place the user's funds in a smart contract that takes \n\nA lambda returning a list of operations\nA counter\nA signature by a given key of the above\n\n(technically you don't really want to explicitly pass the counter, you can just assume the one in your storage is the one that was signed)\nAnd\n\nChecks the signature\nChecks and updates the counter\nExecutes the lambda and return the operations\n\nThis would be very similar to the manager.tz contract except that instead of having the access controlled by checking SENDER, you control the access with a signature and a counter.", "\nNow, have the user produce the lambda and the signature, and any third party can pay a fee to inject this payload into the contract and trigger transactions.", "\n\nA more direct way to do that would be to allow manager operations, which already represent bundles of operations, to involve multiple accounts and multiple signatures so that the fee can be paid by another party. ", "This would be a small, but useful protocol upgrade.", "\n\n" ]

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[ "2016 Tajikistan Cup\n\nThe 2016 Tajik Cup was the 25th edition of the Tajik Cup. ", "The cup winner qualified for the 2017 AFC Cup.", "\n\nLast 16\n\nQuarterfinal\n\nSemifinals\n\nFinal\n\nScorers\n\nReferences\n\nExternal links\nTajikistan Football Federation\n\nCategory:Tajikistan Cup\nTajikistan\nTajik Cup" ]

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[ "import { Row } from 'ember-light-table';\nimport { module, test } from 'qunit';\n\nmodule('Unit | Classes | Row', function() {\n test('create row - default options', function(assert) {\n let row = new Row();\n assert.ok(row);\n assert.equal(row.expanded, false);\n assert.equal(row.selected, false);\n });\n\n test('create row - row instance', function(assert) {\n let row = new Row({ foo: 'bar' });\n let row2 = new Row(row);\n assert.ok(row);\n assert.ok(row2);\n assert.equal(row, row2);\n assert.equal(row.get('foo'), 'bar');\n assert.equal(row2.get('foo'), 'bar');\n });\n\n test('reopen row', function(assert) {\n assert.equal(typeof Row.reopen, 'function', 'reopen is a function');\n assert.equal(typeof Row.reopenClass, 'function', 'reopenClass is a function');\n });\n});\n" ]

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[ "“But not everyone is going to be comfortable pursuing that kind of action and there needs to be other things that they can participate in.”", "\n\n3. ", "The problem with anti-fascists movements" ]

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[ "social\n\nBusinesses who have lived through the evolution of the digital age are well aware that we’ve\nexperienced a generational shift in technology. ", "The rise of software as a service (SaaS),\ncloud, mobile, big data, the Internet of Things (IoT), social media, and other technologies\nhave disrupted industries and changed customers’ expectations. ", "In our always-on, buy\nanything anywhere world, customers want their shopping experiences to be personalized,\ndynamic, and convenient.", "\nAs a result, many businesses are trying to reinvent themselves. ", "Success in a fast-paced\neconomy depends on continually adapting and innovating. ", "Companies have to move quickly\nto keep up; there’s no time for disjointed technologies and old systems that don’t serve the\ncustomer-obsessed mentality needed to thrive in the digital age.", "\n\nData analytic solutions help improve the value of health and human services, drive better decision-making and lower costs. ", "See how the right solutions can contribute to millions of dollars saved through fraudulent claims detection, offer insights into the social determinants of health and benefit social programs.", "\n\nThe creation of an immersive virtual setting requires students to understand the immersive visual and audio environment of the world they intend their characters to inhabit. ", "Development of a digital story arc requires an understanding of sequence, causality, and communication of meaning to digital story viewers. ", "An immersive digital story can be created and viewed on mobile devices, tablets, and computers by students as early as elementary school.", "\nVirtual Digital Storytelling sits at the convergence point of empathy, social and emotional learning, engagement with community, and learning with emerging technologies.", "\nDownload today to find out how Fujitsu's suit of Augmented and Virtual Reality tools can help your business.", "\n\nDid you know that for 88% of shoppers, the first step to a local purchase is often an online search? ", "If you want customers to find you, you must have a consistent, rich and engaging online brand that is equally optimized for both desktop and mobile SEO, and that contains long tail keywords for voice search assistants like Siri, Alexa, Cortana, and Google Voice. ", "Be sure to claim and regularly maintain your listings for every widely used portal and social site, and create a separate locator page for every single location—so that you show up at the top of all local searches.", "\n\nWhether you've had an employee referral program for years, are only in the initial stages of building one, or have never had one at all, this e-book is full of tips and advice on making your program the best it can be.", "\n\nIf you're a brand marketer or digital agency looking to amplify your social marketing initiatives this year, this document will guide you through the growing world of Facebook apps. ", "Today, there are more than 500,000 active applications on Facebook, but of these only 250 applications have more than one million monthly active users. ", "You've downloaded this guide because you recognize that marketing today means creating a two-way dialogue with your customers.", "\n\nB2B demand generation has almost completely re-invented itself over the last decade. ", "The collision of inbound, content marketing, lead nurturing, marketing automation, analytics, email, search, social media and native advertising has transformed the discipline from a dark art to an increasingly data-driven science. ", "But there’s a problem.", "\nWhen it comes to helping to land and expand these major deals, today’s marketing teams are seriously handicapped. ", "The Clear & Complete Guide to Account Based Marketing (ABM) is your guide to solving this problem. ", "It will take you, step-by-step, through this fast-emerging discipline, showing you how to align your sales and marketing team around a strategy that’s optimized for the biggest, most important deals.", "\n\nThe rise of social media is intrinsically connected to email, which was itself the first social network. ", "Email marketers must leverage this complementary channel to expand their reach beyond the email list, realizing the power of viral marketing in the social web.", "\n\nThe growing trend towards insourcing marketing and transactional email is being driven by businesses that are looking for ways to improve their email programs, increase data security and lower costs. ", "When evaluating whether it makes more sense to leverage an on-premise or outsourced solution, it's important to understand how the traditional arguments have changed.", "\n\nWith more than 500 million active users, Facebook can be an extremely effective channel for engaging customers around your brand. ", "However, before you rush out to build a fan page, you need to first understand the best way to interact with them based on conversations about your brand that are already happening on the social web.", "\n\nDoes social media marketing really work? ", "How do you separate fact from fiction as you navigate your way through the buzz about \"going viral\"? ", "Amidst the chatter about Twitter, Facebook, FourSquare, Digg, Delicious, and Linked In-what will really get your business ahead? ", "Tune in to find out as Oneupweb CEO Lisa Wehr and Social Media Director, Maureen Michaels team up to unveil the truth.", "\n\nSocial media is all around us and has fundamentally changed the way many of us communicate--both personally and professionally. ", "Video can make your social media better, and ultimately get more leads to your website. ", "But how best to post, measure, and use to drive traffic to your website?", "\n\nThe hype about video search has just begun which means lots of questions, lots of new experts, and not enough time to sort through it all.", "\nThis new whitepaper \"An Effective Video SEO Strategy in Three Phases\" will give you the blueprint for getting started and beyond.", "\nLearn all about: the 2 things you should be doing right now to improve your video search engine optimization; widespread myths about what works and what doesn't; Advanced tips for even greater results and more!", "\n\nOnly a decade or so ago, those human resources professionals who hadn't yet found their way onto the Internet were finding themselves increasingly left out in the cold.", "As we slip swiftly into the second decade of the 21st century, it's those who haven't yet begun to participate in 'social media and networking' that are starting to feel the chill.", "\n\nOnly a decade or so ago, those human resources professionals who hadn't yet found their way onto the Internet were finding themselves increasingly left out in the cold. ", "As we slip swiftly into the second decade of the 21st century, it's those whohaven't yet begun to participate in 'social media and networking' that are starting to feel the chill.", "\n\nThere’s strong evidence organizations are challenged by the opportunities presented by external information sources such as social media, government trend data, and sensor data from the Internet of Things (IoT). ", "No longer content to use internal databases alone, they see big data resources augmented with external information resources as what they need in order to bring about meaningful change. ", "According to a September 2015 global survey of 251 respondents conducted by Harvard Business Review Analytic Services, 78 percent of organizations agree or strongly agree that within two years the use of externally generated big data will be “transformational.” ", "But there’s work to be done, since only 21 percent of respondents strongly agree that external data has already had a transformational effect on their firms.", "\n\nLearn how ERP software addresses the real-world challenges companies face as they look to grow in today’s global economy.", "\nTake the virtual tour and see how your business can:\n• Reach new markets, expand product lines and grow your business with deep Industry and country specific functionality\n• Drive improved customer experience and accelerate the digital transformation of your business\n• Expand collaboration within your company and throughout your supply chain with social and commerce solutions\n• Respond quickly to changing demands with configurable real-time mobile capabilities\nPlease view the virtual tour below for more information about Epicor ERP version 10. ", "The tour will acquaint you with Epicor ERP and offer insight on how Epicor inspires growth for leading business around the globe.", "\n\nMarketing as you know it will never be the same. ", "There’s a fundamental shift in relationships between brands and customers—fueled by smartphones, social media, and today’s\nalways-on, always-connected mentality. ", "Marketers have access\nto more customer data (big data) than ever before. ", "But the quantity of data only matters if you’re smart about using it—to power 1:1 customer journeys.", "\n\nSAP Jam is a cloud-based social software platform that transforms the way people work by bringing together people, data, and decision-making capabilities when and where workers need them. ", "In this Forrester Total Economic Impact (TEI) study commissioned by SAP, learn about the potential return on investment you can realize by deploying SAP Jam.", "\n\nTo better understand how companies are finding the unique, hybrid cloud architectures that best meet their needs, we interviewed executives at companies that had reduced or changed their use of managed or cloud IaaS or that chose to avoid the public cloud in the first place.", "\nThese companies include retail, social media, healthcare, financial services, and public sector companies. ", "Some of these companies were born in the cloud while others transitioned from traditional IT infrastructures. ", "Company sizes ranged from 300 employees to more than 300,000.", "\n\nWhile the shift from disk to digital offers tremendous potential opportunities, it also presents a host\nof new challenges for gaming companies. ", "As the online channel grows increasingly complex and the pace\nof innovation accelerates, many companies struggle to keep up. ", "Not only are there websites and storefronts\nto manage, but also real-time gaming servers, large software downloads, and live-streamed competitions and\nevents. ", "Games are transforming from fixed, boxed products to dynamic, ongoing services – with frequently\nupdated content, in-game micro-transactions, virtual goods and social interactions. ", "Mobile adds another\ndimension to the trend, as consumers increasingly look to play on smart phones and tablets – or on multiple\nscreens across devices.", "\nTo successfully navigate this complex and changing landscape, gaming companies need an agile,\nhigh- performance infrastructure that allows them to turn the Internet into a reliable and effective\nonline distribution channel. ", "This requires f" ]

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[ "Can standard genotoxicity tests be applied to nanoparticles?", "\nExperiments were conducted to determine the validity of two common genotoxicity testing procedures, the comet assay and the micronucleus (MN) test, when applied to nanoparticles (NP). ", "The comet assay is used to detect strand breaks (SB) induced in cellular DNA. ", "There is a possibility of obtaining false positive results, if residual NP remain in proximity to the virtually naked DNA that results from lysis of agarose-embedded cells, and react with this DNA in ways that do not occur with chromatin in intact cells. ", "However, data showed that if NP are deliberately present at high concentration with lysed cells, there is no change in SB with a range of NP. ", "Only oleic acid-coated Fe₃O₄ NP induced damage, as these particles also produced equivalent alterations in whole cells. ", "A modification of the comet assay incorporates digestion of DNA with lesion-specific endonucleases, notably formamidopyrimidine DNA glycosylase (FPG), which detects oxidized purines. ", "Again there is a concern regarding the presence of residual NP with DNA of lysed cells, but this time because of the risk of false negative results if NP interfere with the FPG reaction. ", "However, it was found that incubation of cells with NP before treatment with a known 8-oxoguanine-inducing agent does not lead to any decrease in the yield of FPG-sensitive sites. ", "Chromosomal damage is detected with the MN assay, which depends on the use of cytochalasin B (CB) to prevent cell division and accumulates binucleate cells. ", "It is known that CB also inhibits endocytosis, and thus might prevent NP uptake. ", "Data demonstrated that if NP are added to cells together with CB, fewer MN are induced. ", "It is therefore necessary to treat cells with NP prior to CB in order to avoid interference and possible false negative results." ]

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[ "The 2018 edition of the European Enterprise Promotion Awards (EEPA) is under way and the search for innovative European projects in the fields of enterprise and entrepreneurship has begun. ", "Continuing the series of testimonials from EEPA 2017, Promoting Enterprise presents the 2017 winner in the category ‘Investing in Entrepreneurial Skills’ – Business Generator from Sweden, represented by Anette Rhudin.", "\n\nHow did you first hear about the national competition?", "\n\nIt was the Swedish Agency for Economic and Regional Growth that called and asked us to apply.", "\n\nWhy did you decide to enter the national competition?", "\n\nOur main motivation for entering was that we wanted more people at the national level to know how we could make difference, and we wanted the national media to write about it. ", "We decided to enter because Business Generator identified a gap in the strategic management process for SME companies, as well as a way to reach SMEs and provide them with useful skills for their daily lives as entrepreneurs. ", "Our intention was to communicate this “gap”, not just in our region, but throughout Sweden. ", "We thought that being National Winners could be a “shortcut” for us, meaning that the Swedish media would address the issue of SMEs and the potential they have in Sweden.", "\n\nHow did you prepare your application?", "\n\nWe spent a lot of time compiling and identifying the reasons behind our project. ", "We wanted to make it easy to understand the complexity of what we do and, of course, to find out what exactly makes us successful.", "\n\nOne of our goals with this project was to actively communicate our results, so preparing the EEPA application was actually very useful and a good way to combine both our communication goal and competition preparation. ", "We interviewed the participating companies and gathered their comments and thoughts about their experience in the project, which was of great help.", "\n\nWhat was it like to win the award?", "\n\nWhen we found out that we were the national winners we hosted a midsummer party, to inform everyone about the prize that we were competing for and to celebrate our national win. ", "It´s in Swedish, but you can see how emotional everyone was, both laughing and crying. ", "It was a great party!", "\n\nWe really wanted to win the European prize, but just because we thought we were the best project didn’t mean that the jury would think the same thing. ", "When Business Generator was announced as the winner, I had a pulse of 1000! ", "Friends who have seen the video of us winning say that I look unmoved, but really I was shocked. ", "Just hearing Sweden mentioned with our project was incredible, I was representing our country!", "\n\nMy own experience of the SME Assembly was overwhelming. ", "There were so many people with the same interests, which gave us a lot of input and a chance to see the differences between countries. ", "This experience gave me a lot to think about and made me realise that Sweden still has a lot of work to do. ", "One evening we talked to representatives from Britain who told us about their own situation and how young entrepreneurs are leaving the UK. ", "This conversation in particular really helped us to understand what the work being done by the EU can mean for all the people in Europe.", "\n\nHow did winning the award impact your work?", "\n\nWhen we became National Winners there were big articles in our local newspapers, but the national papers wrote nothing. ", "After speaking to national radio I learnt that EU issues are rarely covered in Swedish media due to their complexity. ", "This is a shame because our region of Värmland is classified as one of the poorer growth regions in Sweden, meaning it needs some praise and attention in the Swedish media.", "\n\nWhen we won EEPA, social media exploded. ", "It was shared and there was so much gratitude and so many congratulatory messages that we were unable to follow all the threads! ", "In addition, all co-financing municipalities and banks wrote about the win on their websites and social media. ", "Wherever we were, there was always someone telling us how proud they were of our achievement. ", "Even though the Swedish media did not pick up on it as much as we would have hoped, at least people in our sphere seem to really like it and appreciate our efforts.", "\n\nWhy should others enter EEPA 2018? ", "What advice would you give them?", "\n\nThe prize itself is valuable, but so is the opportunity to see how projects in other countries deal with the same issues and questions. ", "You can see differences in financial solutions, project launches and how each country has their own solutions and plans, all of which are the best across Europe.", "\n\nAnother thing to think about is communication. ", "I was so impressed with the communication throughout the SME Assembly! ", "It was really professional and each country was provided with perfect PR. ", "However, there must be media in the home country that receives it, and that is where you need to plan before you go the SME Assembly. ", "We experienced something very extraordinary and I am so grateful. ", "All the people we met, all the information we got, all the big ideas we heard about were so interesting. ", "But if I could do it again, I would have planned more beforehand and talked more to those people that could be useful in the future.", "\n\nIf you like to see how it is possible to change things in a society, then EEPA is a perfect event! ", "I can´t see any better way to be exposed to these kinds of solutions and questions than the EEPA competition.", "\n\nWhat are your plans for the future?", "\n\nThe project Business Generator ended in December 2017 and unfortunately the owner of the project, Inova, ended at the same time. ", "Business Generator was completed as a project, but was far from ready to “fly” on its own. ", "There is still a lot of work to be done in packing, launching and finding public funds in combination with the participating company’s own financing, in order to create a viable Business Generator. ", "We have other programmes in our region, like mentor programmes which are based on people giving their time for free. ", "Our project charged a very low fee for those involved in the Business Generator, meaning that we became a threat rather than an opportunity.", "\n\nWe were hoping that another organisation would take the concept further, but this has not happened yet. ", "In Värmland there are around 7,540 SMEs, all of which need more support and resources, so even though the future of Business Generator is uncertain I hope there will be a way for our project to come back.", "\n\nTo find out more about Business Generator, read about them right here in the 2017 compendium, and be sure to watch their winning moment from the EEPA 2017 ceremony in Tallinnhere.", "\n\nKeep coming back to Promoting Enterprise for more EEPA 2017 testimonials and don’t forget to check all the social media channels (Twitter: @EEPA_EU and Facebook: @PromotingEnterprise) for the latest EEPA updates.", "\n\nThe European Enterprise Promotion Awards (EEPA) are under way across Europe, but what does a competition look like on national level? ", "Today our EEPA journey takes us to Belgium, where the national competition is coordinated by National Coordinator Francis Otte. ", "Since 2006, Belgium has been participating in the EEPA awards scheme, and submitted a great […]\n\nThe EU Startup Monitor Survey is open, and the European Commission wants to hear your voice! ", "The Commission is looking for inputs to create more effective and meaningful policies that support young enterprises who have recently started a business or are in the process of scaling up. ", "Through hearing what Europeans in the sector have […]\n\nThe JADE Spring Conference, the main international Junior Enterprise event in Europe, will take place between March 8th and the 11th, in Brussels. ", "This year’s edition will gather over 400 Junior Entrepreneurs – university students – from more than 14 countries in Europe, as well as Brazil, the United States and Tunisia. ", "After celebrating the […]\n\nIn celebration of the European Enterprise Promotion Awards (EEPA) returning for another edition in 2018, Promoting Enterprise is bringing you a series of testimonials from the EEPA 2017 winners. ", "Today’s winner – Internationalisation 2015-2020 from ecoplus International, shares with us their story of how they became the 2017 winners in the category ‘Supporting the Internationalisation […]\n\nPromoting Enterprise is back with yet another EEPA 2017 winner testimonial for you to be inspired by. ", "Today is the turn of Reempresa, winner of the ‘Improving the business environment’ category with their project for ‘re-entrepreneurs’, represented by director Albert Colomer. ", "How did you first hear about the national competition? ", "We have participated in the […]\n\nEEPA 2018 Launch It is that time of the year again, when the European Enterprise Promotion Awards (EEPA) commence and the best projects from all across Europe begin to compete for a place on the EEPA 2018 shortlist. ", "Today 05 February, marks the opening of the EEPA 2018 and the project nominations, which will be happening […]\n\nAll the way from the sunny Portuguese island of Madeira, Promoting Enterprise is bringing you a story from a former European Enterprise Promotion Awards (EEPA) nominee…Madeira Vintners! ", "Over the last five years, the all-female team at MADEIRA VINTNERS have been working to establish their Madeira wine brand, the first Madeira wine firm to be founded […]\n\nThe 2018 edition of the European Enterprise Promotion Awards (EEPA) is under way and the search for innovative European projects in the fields of enterprise and entrepreneurship has begun. ", "Continuing the series of testimonials from EEPA 2017, Promoting Enterprise presents the 2017 winner in the category ‘Investing in Entrepreneurial Skills’ – Business Generator from Sweden, […]\n\nThe Entrepreneurship Conference “Follow your ideas – become an entrepreneur” will take place in Sofia, Bulgaria on 6 March 2018 under the Bulgarian presidency of the Council of the European Union. ", "The conference is organised by the Ministry of Economy of the Republic of Bulgaria and the European Commission. ", "The conference will bring together participants […]\n\nThe European Enterprise Promotion Awards (EEPA) are back again for 2018, and ready to discover more outstanding European projects in the fields of enterprise and entrepreneurship. ", "In order to help potential applicants, Promoting Enterprise has asked EEPA 2017 winners to share their success stories give advice on how to submit and present an EEPA winning […]" ]

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[ "2k SHARES Facebook Twitter Whatsapp Pinterest Reddit Print Mail Flipboard\n\nAdvertisements\n\nThe latest in Donald Trump’s long sequence of impeachable offenses just came to light this morning.", "\n\nIn a new report from The New Yorker’s Jane Mayer it is revealed that Trump tried to use the Department of Justice (DOJ) to punish CNN for their unfavorable coverage of him.", "\n\nTrump ordered former economic adviser Gary Cohn to put pressure on the DOJ to block a proposed merger between AT&T and the parent company of CNN, Time Warner. ", "It was a massive business deal that would have benefit both Time Warner and CNN. ", "Cohn resigned one year ago, deciding he couldn’t take it anymore.", "\n\nAdvertisements\n\nAccording to Mayer, Trump dragged Cohn into the Oval Office in the summer of 2017 and verbally abused him for not doing enough to block the proposed huge business deal. ", "Mayer wrote:\n\n“According to a well-informed source, Trump called Cohn into the Oval Office along with John Kelly, who had just become the chief of staff, and said in exasperation to Kelly, “I’ve been telling Cohn to get this lawsuit filed and nothing’s happened! ", "I’ve mentioned it fifty times. ", "And nothing’s happened. ", "I want to make sure it’s filed. ", "I want that deal blocked!”", "\n\n“Cohn, a former president of Goldman Sachs, evidently understood that it would be highly improper for a President to use the Justice Department to undermine two of the most powerful companies in the country as punishment for unfavorable news coverage, and as a reward for a competing news organization that boosted him. ", "According to the source, as Cohn walked out of the meeting he told Kelly, “Don’t you fucking dare call the Justice Department. ", "We are not going to do business that way.”", "\n\nThe DOJ would later file a lawsuit to block the merger months later, Mayer wrote, even though Cohn also apparently disobeyed Trump’s orders to get involved in the deal.", "\n\nAccording to Mayer, while Cohn and Kelly did not comment on the anecdote she wrote about, a former White House official confirmed that the president frequently “vented” in “frustration” about the deal. ", "The official said that Trump ranted regularly about DOJ inaction over the AT&T-Time Warner merger.", "\n\n“The President does not understand the nuances of antitrust law or policy,” the official explained. “", "But he wanted to bring down the hammer.”", "\n\nGeorge Conway, the husband of Trump White House adviser Kellyanne Conway, wrote on Twitter that Trump should be impeached if it’s found that Mayer’s reporting is correct.", "\n\n“If proven, such an attempt to use presidential authority to seek retribution for the exercise of First Amendment rights would unquestionably be grounds for impeachment,” Conway tweeted in response to Mayer’s report.", "\n\nIf proven, such an attempt to use presidential authority to seek retribution for the exercise of First Amendment rights would unquestionably be grounds for impeachment. ", "https://t.co/F1UANzeD2q — George Conway (@gtconway3d) March 4, 2019\n\nThis is true, but Louisiana Republican Senator John Kennedy said it was fine with him if Trump blocked the merger of AT&T-Time Warner because he didn’t like the way he’s covered by CNN. ", "Kennedy said that Trump, like anyone else, has the right to an opinion.", "\n\nRepublicans in Congress, as always, are covering up for Donald Trump and enabling him in the commission of crimes and impeachable offenses." ]

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[ "The present invention relates generally to an apparatus used for aiding in the transport of payloads from the Earth\"\"s surface to outer space. ", "Specifically, the present invention relates to a reusable flyback booster that incorporates a removable rocket propulsion module within an airframe. ", "Still more specifically, the present invention relates to a reusable flyback booster comprising an aircraft, having air breathing engines and capable of landing on a runway, said aircraft functionally enclosing a separate launch vehicle stage. ", "The present invention may be used as the first stage of a multistage launch vehicle. ", "Thus the present invention also relates to a comprehensive launch vehicle system architecture wherein the stage used in the flyback booster and the upper stages are selected to cost optimize the multistage launch vehicle for launch of a specific payload or class of payloads.", "\nThe present invention\"\"s background art is generally found in the art of space launch vehicles. ", "Patents in this field of art are classified generally in international class B64G and in U.S. class 244.", "\nThe inventors identify the following background art because they believe it will be useful in understanding, searching, and examining the invention. ", "This invention lies in an area of the technical arts where it really is necessary to be a rocket scientist to understand the invention. ", "The inventors anticipate that they may be required to explain the invention to persons who are not technically trained, such as administrators, jurists or judges. ", "To aid in understanding the present inventors they offer: 1. ", "an overview of information on space launch vehicles, including some technical and cost data on certain specific vehicles; 2. ", "information on the background art related to partially reusable launch systems and 3. ", "relevant U.S. patents. ", "Taken as a whole, this body of information represents the current state of the art in launch vehicle recoverable boosters, as the inventors know it at the time of their filing of this patent application. ", "It also illustrates the enormous complexity of this field of art, which is currently in a dynamic period of development.", "\nAlthough sounding rockets may reach altitudes above the atmosphere of the Earth, the term space launch vehicle is applied usually to those rocket boosters designed to place satellites in orbit or to impart Earth-escape velocity to spacecraft.", "\nBy about 1950 the technology of rocket propulsion had reached a level at which consideration of a project to launch an Earth satellite became feasible. ", "Worldwide scientific studies during the IGY of 1957-58 provided the basis for funding. ", "In 1955 both the United States and the Soviet Union announced satellite programs as part of their national effort in the IGY.", "\nWhen Sputnik 1 and 2 were launched in 1957 the Soviet Union released no details of their launch vehicles. ", "In May 1958 Sputnik 3, weighing nearly 1,360 kilograms, was launched. ", "It was not until 1967 that the basic Soviet launch vehicle was displayed. ", "It was a 2xc2xd-stage vehicle of the xe2x80x9cAxe2x80x9d series (in this case, xe2x80x9cA-1xe2x80x9d): two stages with four drop-away booster pods. ", "Each booster pod contained four rocket engines (totaling 16) with propellant tankage, and the central core had four engines. ", "Propellants were liquid oxygen and kerosene.", "\nThe United States launched its early satellites with two different vehicles, the Jupiter-C and Vanguard. ", "Jupiter-C was a modified Redstone liquid-propellant ballistic weapon of medium range to which were added more tankage length and three upper stages of clustered solid-propellant rockets. ", "The modification was originally designed to achieve a velocity of six kilometers per second to test a nose cone (reentry vehicle). ", "The desired velocity was obtained with two upper stages, one a cluster of four solid-propellant rockets and the other a single rocket. ", "By increasing the final velocity 1.5 kilometers per second to the required 7.5 kilometers per second, satellite velocity could be obtained for a small scientific payload. ", "The additional velocity was obtained by adding another stage with a cluster of solid-propellant rockets so that the upper stages consisted of 11, three, and finally one rocket carrying a payload weighing 8.2 kilograms. ", "In 1954 the Army Ballistic Missile Agency and the Office of Naval Research jointly proposed this scheme, known as Project Orbiter, but a newly designed Vanguard launch vehicle was selected. ", "Failures in early attempts to launch Vanguard, however, resulted in eventual approval of the Project Orbiter approach. ", "Thus the first U.S. satellite, Explorer 1, was launched by a Jupiter-C on Jan. 31, 1958.", "\nThe Vanguard launch vehicle was a three-stage booster approximately equal in length (about 22 meters) to the Jupiter-C but much lighter in takeoff weight (10,250 kilograms compared to 29,000 kilograms). ", "Vanguard launched its first satellite (1.4 kilograms) into high orbit on Mar. 17. ", "1958. ", "After a few more flights, the Jupiter-C was retired in 1958 and the Vanguard in 1959.", "\nDuring the 1960s the United States developed a series of standard launch vehicles. ", "The Air Force modified a Titan II intercontinental ballistic missile (ICBM) for space launch purposes by strapping two solid-propellant booster rockets, three meters in diameter, to the liquid-propellant core vehicle. ", "The Titan IIIC was used for large military satellites. ", "Then NASA increased performance of the obsolete Thor intermediate-range ballistic missile (IRBM) by adding solid-propellant boosters. ", "A liquid oxygen/liquid hydrogen upper stage, Centaur, was used on obsolete Atlas and Titan ICBMs to launch large spacecraft.", "\nThe Saturn series of NASA launch vehicles was developed specifically for the Apollo lunar mission program. ", "The two operational Saturn models were the two-stage Saturn IB and three-stage Saturn V. The Saturn IB was used for Earth orbital developmental missions of Apollo, while the Saturn V was employed for lunar missions. ", "Saturn V stood 110.6 meters high and weighed over 2,700,000 kilograms at launch. ", "It could place 104,000 kilograms in orbit and send 45,000 kilograms to escape velocity.", "\nFor some years the launching of spacecraft was limited to the United States and the Soviet Union. ", "The reason was that the early rocket-powered launch vehicles were based on long-range ballistic missiles, which only these countries had developed. ", "France was the third nation to launch a satellite (1965), followed by Japan (1970), the People\"\"s Republic of China (1970), and the United Kingdom (1971). ", "Under the auspices of the European Space Agency (ESA), the nations of Western Europe developed the Ariane expendable launcher during the 1970s to assure themselves of independent launch capability. ", "This action was taken, in part, in response to the U.S. refusal to guarantee flights for communications satellites that might compete with U.S. telecommunications carriers. ", "A three-stage vehicle that burns storable and solid propellants in its first two stages and employs a cryogenic engine in its third, Ariane has become a formidable competitor for commercial space launch services, capturing about half of the global market. ", "It is capable of launching two satellites of the U.S. Delta class (an Earth-orbit payload of 1,770 kilograms) at one time or one Atlas-Centaur-class satellite (an Earth-orbit payload of 4,670 kilograms). ", "With the new cryogenic propellant core, Ariane is approaching payload weights that only the Shuttle can handle.", "\nAn important factor affecting space mission cost is the cost of launch vehicles. ", "Many of the major launch vehicles are designed to place payloads of 1500-6000 kg into geostationary transfer orbit, at a typical cost of $50M-150M. Such launch performance is to little and costs are far too high for the low-cost missions satellite operators require.", "\nMany of the early Western space launchers had, by modern standards, very small payloads. ", "The U.S. Vanguard rocket could place 20 kg into LEO, whilst the French Diamant and British Black Arrow both had LEO capabilities of approx 50 kg. ", "From the mid-1960s onwards launcher development concentrated on increasing payload weight, but during the 1980s the growing interest in small satellite development lead to a number of small and (relatively) low-cost launchers being produced. ", "A further source of small launchers opened up in the late 1980s with the decommissioning of many U.S. and former Soviet nuclear missiles. ", "Russia in particular has been keen to convert former military missiles into launchers which can be sold for foreign currency.", "\nUnless otherwise specified LEO (Low Earth Orbit) and polar orbit payload data are for a 100 nm orbit. ", "LEO performance is generally given for the lowest inclination achievable from the vehicle\"\"s main launch site. ", "In some cases, sources provide performance data for non-standard orbits without explicitly saying so. ", "This can introduce some errors into the data for less common vehicles.", "\nGTO stands for Geostationary Transfer Orbit, and should not be confused with GEO, Geostationary Earth Orbit. ", "The satellite or an attached apogee kick motor generally performs the impulse from GTO to GEO, so launch vehicles often specify only GTO capability.", "\nPrice and performance data may vary. ", "Launch prices depend on the spacecraft, currency exchange rates, and market fluctuation. ", "Payload depends on fairing and adapter selection. ", "This data should be accurate enough to make comparisons and conduct preliminary analysis.", "\nIt is difficult to find comprehensive data for some Russian or Chinese systems since they were often secret, and data on the more obscure foreign launch systems doesn\"\"t get published very frequently. ", "When data is available, sources sometimes disagree. ", "Therefore, reliability data for a few launchers may be out of date or inaccurate.", "\nThe Ariane 4 series holds the largest market share in the international commercial launch market. ", "Development is funded by the European Space Agency and lead by CNES, the French space agency. ", "Arianespace conducts operations. ", "The vehicles launch from French Guiana in South America. ", "Ariane 5 was designed to launch multiple large communications satellites for a lower cost than previous versions. ", "However, satellites have continued to grow since the program was started almost ten years ago. ", "There is speculation that Ariane 5 will eventually be too small to launch two satellites, but too large to launch just one. ", "Therefore, ESA has approved a roughly $1-2 billion xe2x80x9cAriane 5 Evolutionxe2x80x9d project to increase GTO payload to about 7.4 tons in small increments after to the year 2000.", "\nAtlas is the largest commercial launch vehicle in the U.S. and is used frequently for commercial and military launches. ", "Starting in the summer of 1995, Atlas was marketed jointly with the Russian Proton vehicle by International Launch Services, a joint venture of Lockheed Martin and Russian aerospace companies. ", "This offers more flexibility for customers.", "\nThe Delta launch vehicle family is built and marketed by McDonnell Douglas (now Boeing). ", "The Delta II has proved reliable, but is too small for most geosynchronous satellites. ", "Therefore, McDonnell Douglas is developing the Delta III, with a much larger payload. ", "Hughes has purchased 10 launches for its satellites. ", "New Delta versions were also designed for NASA\"\"s Med-Lite contract, which sought launch vehicles between the size of small launchers like Pegasus, and the Delta II, which was the smallest of the large launchers. ", "The smaller Delta versions will be used for future Mars missions, among other things\nThe H-2 is the first Japanese launch vehicle to be entirely developed domestically. ", "Previous N series and H-1 vehicles used Delta components. ", "The H-2 is designed to carry heavy payloads to orbit and has worked well so far. ", "However, it is unlikely to be commercially attractive in the near future, due to high costs and low flight rates. ", "NASDA hopes to cut costs by as much as 50% by the turn of the century, in part by simplifying the design and including some foreign components. ", "The H-2 is the cornerstone of NASDA\"\"s plans for increasing activities in space, including eventual human missions.", "\nKosmos (also spelled Cosmos) is a Russian vehicle comparable in size to the American OSC Taurus launch vehicle. ", "Following back to back failures of the Pegasus XL, LLV. ", "and Conestoga in the summer and fall of 1995, Kosmos attracted attention in the United States as an alternative launcher with a more reliable history. ", "Several companies have worked out joint agreements with the manufacturer, Polyot. ", "Assured Space Access appears to be the current favorite, although other companies have also been involved. ", "Final Analysis Inc. has reserved a number of launches for its own use and is marketing extra payload space on those launches. ", "Kosmos has reached orbit at least 389 times.", "\nThe first flight of the LLV-1 (now called Athina 1) failed during the summer of 1995 when the vehicle began pitching out of control. ", "Fortunately, the vehicle had a good order book for such a new vehicle, including NASA\"\"s Lewis and Clark satellites, and the Lunar Prospector mission. ", "Therefore the LLV overcame this initial setback.", "\nPegasus was the first new American vehicle in more than a decade, and deserves some credit for restarting the interest in small satellites. ", "Pegasus is a small, all solid rocket vehicle built by Orbital Sciences Corporation. ", "The winged rocket is launched from beneath the company\"\"s L1011 aircraft. ", "The original Pegasus configuration is being phased out, in favor of the Pegasus XL (Extended Length). ", "The first two Pegasus XL flights were failures.", "\nTaurus was developed to meet military requirements for rapid launch of small spacecraft. ", "It consists of Pegasus stages mounted atop a Castor 120 first stage.", "\nProton is the heavy lift workhorse of the former Soviet launch stable. ", "It is being marketed in the west by International Launch Services, a joint venture between Krunichev and Lockheed Martin. ", "ILS also offers the Atlas. ", "Russia is currently limited to offering prices within 7.5% of western prices and the number of GEO launches is limited to 8 before the year 2000. ", "However, there is speculation that these restrictions may be abandoned as Russian launches become more commercialized. ", "ILS has twelve western contracts for Proton launches, starting in 1996 with an Astra satellite for Societe Europeenne de Satellites of Luxembourg. ", "Proton is also scheduled to play an important role in launching space station components. ", "Krunichev plans to offer new upper stages for Proton, including the storable propellant Breeze-M upper stage in 1998 and the OHSM cryogenic stage a few years later. ", "Proton will put 3.2 tons in GEO with Breeze-M and 4.5 tons with OHSM. ", "Current GEO capability is about 2.6 tons with the Block D upper stage. ", "In addition to these technical changes. ", "ILS is considering conducting Proton launches from Cape Canaveral, or sites in Australia or Brazil. ", "Launching closer to the equator would increase performance.", "\nShavit is Israel\"\"s first, and so far only, launch vehicle. ", "It is derived from the Jericho II ballistic missile. ", "Israel Aircraft Industries is developing a more advanced version with an added stage, which would be called xe2x80x9cNext.xe2x80x9d The payload of the new vehicle would be slightly higher than Pegasus, and a cost of $15 million has been suggested. ", "Commercialization is desired because Israeli missions number less than one a year and have limited government support. ", "In order to avoid dropping spent stages on Arab neighbors, Israel launches west over the Mediterranean, decreasing the vehicle\"\"s performance significantly.", "\nTitan II vehicles are left over ballistic missiles that have been refurbished for space launch. ", "They are used for polar orbiting Earth observation systems. ", "It was a Titan II that launched Clementine. ", "Titan IV is used mainly for large military payloads, including Milstar communications spacecraft and classified intelligence platforms. ", "A Titan IV is also booked to launch NASA\"\"s Cassini mission to Saturn. ", "Note that because all Titan IV launches are government missions, and most are classified, prices are subject to debate.", "\nZenit is the newest of the large former Soviet vehicles, having come online in 1985. ", "It suffered three consecutive failures between 1990 and 1992. ", "NPO Yuznoye manufactures Zenits in Ukraine. ", "Boeing has a joint venture with NPO Yuznoye and the Norwegian marine engineering company Kvaemer to launch Zenits from a modified oil platform. ", "Due to the lower launch site latitude and a new upper stage from RSC Energia, performance will increase. ", "Payload to GTO will increase to about 5400 kg. ", "Payload to LEO will be about 13.000 kg.", "\nX-34 (United States)\nX-34 is a semi-reusable vehicle. ", "Its development was funded in part by a $70 million contract with NASA. ", "OSC, which spent a total of $100 million on the project, manage the project. ", "Plans are for the vehicle to be carried atop a NASA 747 shuttle transporter and launched at altitude. ", "The vehicle would reach roughly half of orbital velocity and eject a satellite with an expendable upper stage to reach orbit. ", "Estimated price is around $4 million per launch.", "\nEELVxe2x80x94Evolved Expendable Launch Vehicle (United States)\nThe U.S. Air Force has the responsibility for funding development of U.S. government ELV programs; EELV is their answer. ", "The Air Force has provided about $1 billion to Boeing and Lockheed Martin each of them to develop a new launch vehicle that can launch all military spacecraft. ", "Theoretically, the consolidation would mean high flight rates for these two launch vehicle types, thus lowering unit costs. ", "These two vehicles are currently under active development.", "\nA number of small space launch vehicles that could be used with the present invention are in planning and development. ", "These include:\nESA/CNES Small Launchers (Europe)\nESA and the French space agency CNES have considered all manner of small launchers, be they solid or liquid, air launched or ground launched. ", "Proposals have included derivatives of Ariane, various national missile programs, or Russian vehicles like Soyuz. ", "The current study project is the European Small Launcher (ESL), an all solid vehicle which could launch one ton into a 700 km sun-synchronous orbit for $20 million.", "\nItalian Small Launchers (Italy)\nA variety of small launch vehicles have been studied and tested by the Italian space agency, University of Rome, and Italian aerospace firms. ", "Generally the vehicles are derived from Scout components, since Italy has experience launching Scout rockets from their San Marco platform off the coast of Kenya. ", "Various projects have gone by the names Vega, Zefiro, San Marco Scout, Advanced Scout, etc.", "\nKistler K-1 (United States)\nKistler is an aerospace company, which is using private funds to develop an all reusable, two stage small launch vehicle. ", "Tests of hardware for the K-0, a subscale engineering test vehicle, have been conducted. ", "The Kistler fleet would include the K-1, with a payload of 2000 pounds to LEO starting around the turn of the century, and the K-2 which would carry 6000 pounds a starting a few years later. ", "Eventually, Kistler would like to build the K-3, which could launch 20,000 lbs. ", "The company is releasing little public information, and management and engineering shakeups have been occurring, which could affect the design and timeline for the fleet. ", "For more information, see the Kistler homepage at http://www.newspace.com/Industry/Kistler/home.html.", "\nPacAstro (United States)\nPacAstro now has at least three contracts; customers include KITcom of Australia which plans to launch satellites similar to Orbcomm, and the Swedish Space Corp. Much of the technology will be developed under contract with U.S. Air Force for a sounding rocket dubbed PA-X. The PA-2 would carry 340 kg (750 lbs) to LEO or 225 kg (500 lbs) to a polar orbit for $6 million dollars\nRockot (Russia/Germany)\nRokot is a three stage liquid propellant launch system developed in Russia and funded in part by German companies. ", "Eurockot Launch Services GmbH will market it. ", "Rockot is derived from the SS-19 ICBM with an additional upper stage, and should be able to put about 1800 kg into low orbits.", "\nRussian Small Launchers (Russia):\nA large number of new small launch vehicles are being designed in Russia. ", "They are usually derived from ICBMs or SLBMs. ", "They include:\nRiksha-1: Under development at NPO Energomash, to launch 1.7 tons to LEO for $10 million.", "\nSurf: Sea-launched vehicle derived from the SSN-23 and SSN-20 submarine ballistic missiles.", "\nSpace Clipper: Air launched version of SS-24. ", "This is a Ukraine venture. ", "The manufacturer is NPO Yuzhone.", "\nSeagull (Russia/Australia)\nRussian organizations and the Australian Space Office are discussing a project to co-produce a liquid-fueled space launcher with a capacity of about one ton into low orbit. ", "The vehicle would be a new design, though it would use a number of existing components. ", "Launch would take place either from Woomera or a site on the northeast coast of Australia.", "\nVLS (Brazil)\nThe VLS has been a long standing goal of the Agencia Espacial Brasileria and a major part of the Brazilian Complete Space Misson (MECB). ", "The launcher is derived from the Sonda IV sounding rocket and is currently designed to put 185 kg into a 750 km orbit.", "\nReusable Launchers\nA frequently-proposed solution to the high cost of access to space is the development of reusable launch vehicles. ", "It has often been pointed out that if aircraft were built for every flight and scrapped thereafter, then air travel would be as expensive as space launch. ", "Unfortunately the economic as well as technical challenges involved in building a reusable launch vehicle (RLV) are considerable, as is evident from the continued use of expendable launchers some 40 years after the launch of the first artificial satellite. ", "The one operational launch vehicle with any degree of reusability the U.S. Space Shuttle, is as expensive to operate in terms of $/kg to LEO as most expendable launchers and requires a large support operation to refurbish each Orbiter after flight. ", "The main difficulty with building a fully reusable launch vehicle, particularly one which is a single unit, is that of achieving orbital velocity (9,500 m/s including drag and gravity losses) without using disposable stages or tanks.", "\nxe2x80x9cIspxe2x80x9d the abbreviation for is a quantity known as the specific impulse of the particular propellant combination and rocket engine being used. ", "Specific impulse is measured in units of pounds force per pounds mass per second, frequently shortened to xe2x80x9csecondsxe2x80x9d. ", "Commonly used propellants have values of Isp between 260s and 450s; to achieve a final velocity of 9,100 m/s with them would require a mass ratio of 8 to 35. ", "Mass ratios much above 7.3 are very difficult to achieve, especially with any significant payload, using those propellants with sufficiently high Isp to allow lower mass ratios: liquid hydrogen as a fuel is low-density, leading to difficulty in building large tankage while maintaining the required mass ratio. ", "Also, to maintain a reasonable level of acceleration at lift off, a launcher must carry heavy rocket engines that significantly vary its thrust during its burn. ", "Expendable launchers drop off structure such as empty tanks and the large engines needed for takeoff, thus maintaining their effective mass ratio and lowering thrust throughout flight. ", "Building an RLV as a single unit is thus an exceedingly difficult proposition.", "\nIn recent years, advances in materials technology have given rise to the hope that it may be barely possible to build a single-stage-to-orbit (SSTO) RLV. ", "By using strong but light materials such as composites and advanced alloys it may be possible to build a vehicle light enough in relation to its size that it\"\"s mass ratio is high enough to achieve orbit with a useful payload. ", "The NASA/Lockheed-Martin X-33 is being built as a technology demonstrator for such a vehicle, and should carry out suborbital flights in late 1999 or 2000. ", "It is hoped by NASA that such work will lead to the commercial development of operational SSTO-RLVs within the next 10 years. ", "An objective of such research is to develop an RLV that can be refurbished between flights with little more than the routine servicing required by conventional aircraft. ", "Progress was made in this area by the McDonnell-Douglas DC-X/XA before it was lost in a flight accident, with successful demonstration of rapid turnaround by a small flight crew. ", "If it is successful, this research could lead to an RLV that can fly at short notice (under a week) to deliver to orbit a payload at relatively low cost (under $1000/kg to LEO). ", "Such a vehicle would be ideal as a launcher for rapid-response missions, as it would make it feasible to procure a dedicated launch into a specific parking orbit at short notice and at reasonable cost. ", "However, as mentioned, it is unlikely that such a vehicle will enter operational service before 2007, with 2010-2015 or beyond being more likely. ", "As such, whilst the launch of rapid-response missions is likely to become a much easier proposition with the development of SSTO-RLVs, this will not happen in the near future. ", "Very low payload fractions and high risk and development costs of SSTO vehicles make many experts believe that they will never by a cost-effective means of launching significant payloads.", "\nHowever, the SSTO-RLV is not the only approach to low-cost space access. ", "Making any element of an ELV reusable should reduce costs. ", "This is only true, however, so long as the element in question actually is easy to refurbish, and the cost of developing such a capability is not excessive. ", "The U.S. Space Shuttle, for instance, reuses all elements except its external fuel tank. ", "However, the high development cost and the effort involved in refurbishment between flights results in its cost to orbit being as high as most expendable launchers. ", "Such savings are likely to be more pronounced if the part reused is the largest component, i.e. the first stage. ", "The first stage is a particularly promising candidate for replacement, as not only does it have to carry the payload and upper stages, but it must overcome the most significant delta-V overheads (atmospheric/gravity drag) associated with launch. ", "These overheads typically add 150-2,000 m/s to the delta-V required to reach normal LEO orbital velocity of 7,800 m/s. Even small improvements in this area can bring dramatic benefits. ", "A good example is the Orbital Sciences Corporation (OSC) Pegasus air-launched rocket, which uses a converted L-1011 airliner as a reusable xe2x80x98zeroth stagexe2x80x99. ", "Even by taking the remaining stages to just 9,000 meters altitude and 250 m/s velocity, the required mass of a Pegasus rocket is approximately halved in takeoff weight from a ground-launched version of equivalent performance. ", "An additional factor in making just the first stage reusable is that its recovery is much easier than for an SSTO. ", "An SSTO returning from orbit must cope with re-entry at approximately Mach 25, whereas a reusable first stage would reach a peak velocity of Mach 3 to Mach 6.", "\nA number of partly reusable launch vehicles (PRLV) have been proposed around the concept of a reusable first stagexe2x80x94often air-launched in some wayxe2x80x94being used to boost the payload and upper stages onto a suborbital trajectory. ", "During 1995 OSC carried out development work on the X-34, effectively a reusable vehicle that replaced the first stage of Pegasus. ", "Although the X-34 has been redesigned as a suborbital test vehicle, other companies have made similar proposals. ", "Kelly Space has begun development work on the Eclipse winged launcher that would be towed behind an airliner before climbing on rocket power into a trajectory where solid-propellant upper stages and the payload would be released from a cargo bay. ", "Pioneer Rocketplane\"\"s xe2x80x98Pathfinderxe2x80x99 (originated as the U.S. Air Force xe2x80x98Black Horsexe2x80x99 spaceplane study) is a small spaceplane that enhances its performance via aerial propellant transfer. ", "It takes off on a runway using two conventional augmented turbofan engines, carrying a full load of kerosene fuel but only sufficient liquid oxygen (LOX) oxidizer to accomplish the required propellant transfer. ", "It rendezvouses with a tanker aircraft to take on a full load of LOX before using its LOX/kerosene rocket engine to climb to 100 km altitude at a speed of Mach 12. ", "Here it releases a solid-propellant upper stage (typically a STAR48 or -63 boost motor), which accelerates the attached payload into orbit. ", "The Pathfinder then re-enters and relights its turbofan engines to return to base. ", "Aerial propellant transfer allows the Pathfinder to reduce structural weight as its wings and landing gear are sized to support a much lower weight than if it took off with a full propellant load. ", "Speculations indicate that Pathfinder should be able to place 1,000 kg into polar LEO, or more into orbits of lower inclination, for an estimated cost of $5M. Pioneer is currently carrying out a design study on the Pathfinder for NASA as a contender for the Bantam-X RLV proposal; it is also a contender for the proposed USAF space sortie vehicle.", "\nNASA is studying the possibility of using liquid fueled fly back boosters (xe2x80x9cLFFBxe2x80x9d) as part of the U.S. National Space Transportation System, commonly called the xe2x80x9cSpace Shuttle.xe2x80x9d The LFBB is designed to be a completely reusable liquid-fueled booster that will return to the launch site, using an autonomous landing system, after separation from the Orbiter and external tank. ", "By way of contrast, the current Shuttle SRBs have to be recovered from the ocean and then extensively refurbished for later use. ", "The initial proposal for the LFBBs includes two options: xe2x80x9cdualxe2x80x9d boostersxe2x80x94two separate winged boosters; and xe2x80x9ccatamaranxe2x80x9d boostersxe2x80x94a dual fuselage connected by a common wing. ", "The catamaran version has been abandoned for technical reasons. ", "Wind tunnels tests of the LFBB shuttle configuration have been conducted. ", "Both Boeing and Lockheed Martin have performed concept definition studies of the LFFB. ", "This work is continuing.", "\n1. ", "xe2x80x9cInternational Reference Guide to Space Launch Systemsxe2x80x9d by Steven J. Isakowitz, 1991 edition. ", "Published by AIAA.", "\n2. ", "xe2x80x9cTransportation Systems Data Bookxe2x80x9d NASA Marshall SFC. ", "Revision A 1995.", "\n3. ", "xe2x80x9c1991-1992 Europe and Asia in Space,xe2x80x9d compiled by Nicholas Johnson and David Rodvold for USAF Phillips Lab.", "\n4. ", "As an additional source of information, NASA maintains a web page at http://www.ksc.nasa.gov/elv/elvpage.html which includes some information about expendable launch vehicles used by NASA.", "\n5. ", "xe2x80x9cCapabilities, Costs, and Constraints of Space Transportation for Planetary Missions,xe2x80x9d by Karen Poniatowski and Michael Osmolovsky of NASA HQ\"\"s Launch Vehicle Office. ", "This paper, along with papers on planetary capabilities of the Delta, Titan II and M-V were presented at the 1994 IAA International Conference on Low-Cost Planetary Missions, and are archived in Acta Astronautica, Vol. ", "35, 1995.", "\n6. ", "More recent information on launch vehicles may be found at www.jsc.nasa.gov/bu2/launch.html\nU.S. Pat. ", "No. ", "3,702,688 teaches a fully recoverable launch system consisting of two piloted winged vehicles that launched vertically in a mated configuration. ", "A first larger booster vehicle was to have propelled the mated pair to an altitude of about 65 kilometers where the second vehicle was to be released. ", "This system was never built.", "\nU.S. Pat. ", "No. ", "3,929,306 teaches a partially recoverable space shuttle system with rocket engines carried below an external fuel tank. ", "The engines are recovered by loading them on board the Orbiter after their fuel is exhausted.", "\nU.S. Pat. ", "No. ", "4,557,444 teaches a dual structure SSTO aerospace vehicle having an aero shell structure and an internally disposed separable and reusable integral tank for its hydrogen fuel and its liquid oxygen.", "\nU.S. Pat. ", "No. ", "4,796,839 teaches a launch vehicle whose first stage rocket motors are ejected and recovered after first stage burnout when they by use of aeroshells and parachutes.", "\nU.S. Pat. ", "No. ", "4,834,324 teaches a space transportation system comprising several different modular units usable in a plurality of different configurations for different payloads and space missions. ", "All units include aerodynamic devices such as fixed or swing-wings for a controlled returned to a runway landing on Earth.", "\nU.S. Pat. ", "No. ", "5,143,327 teaches a heavy lift launch vehicle having disposable fuel tanks and a plurality of winged flyback propulsion modules as a first stage.", "\nU.S. Pat. ", "No. ", "5,402,965 teaches the use of a carrier aircraft to fly a launch vehicle to a launch altitude and velocity. ", "The significant teaching of this patent is the use of a detachable positioning apparatus that is recovered and reused from the launch vehicle.", "\nU.S. Pat. ", "No. ", "5,740,985 teaches a two-stage launch system, the first stage consisting of two flyback boosters or aircraft.", "\nHaving studied all of the prior art, the present inventors draw the following conclusions:\nFirst, the main impediment to the development of space industry is not technical or engineering problems. ", "It is purely economic. ", "The high cost (now above $5,000 to $20,000 per kilogram) of launching payloads to space precludes significant development of a space industry, space tourism and the all other development of space as an ordinary venue for business and personal activities. ", "Many econometric studies have shown that a cost reduction of at least one order of magnitude, to less than $1,000 per kilogram, will be required to make space a practical place to live and work.", "\nSecond, the greatest economic savings should come from the lowering of the costs of operating the first stage of any launch vehicle. ", "One way to do this is to make the first stage booster reuseable. ", "Hundreds of millions of dollars have been spent and thousands of engineers have worked for years on this problem. ", "The original NASA proposal for the Space Shuttle included a fully reusable first stage, but was not accepted due to high development costs. ", "This work, as, for example, on the LFBB, is continuing. ", "Mountains of paper studies have been written on the problem. ", "All of them require the development of entirely new rocket propulsion systems. ", "Such novel systems are extremely expensive to design and test. ", "The development costs of such systems have been found by past studies to largely negate the cost savings realized by the reusable booster.", "\nThird, nothing in the prior art known to the present inventors teaches the use of an inexpensive existing booster stage as a removable propulsion module an aircraft that can function as a flyback first stage booster in a multistage launch vehicle system.", "\nThe present invention is a flyback booster comprising an airplane housing a separable rocket propulsion module. ", "The propulsion module may be a modified existing rocket stage, such as the Zenit, Atlas III or others; or it may be a newly designed and built rocket propulsion module.", "\nThe advantages of the present invention include separation, to the maximum extent possible, of the development, procurement and maintenance of the aircraft and rocket propulsion modules. ", "This separation is expected to result in major economies in the procurement and operations of the launch system.", "\nThe removable rocket propulsion module may be lightly constructed because the airplane fuselage carries all aerodynamic and bending loads, transmits thrust to the payload and provides protection from externally generated heat and sound pressure levels. ", "Attachment of the module to the fuselage and transmission of thrust loads from the module to the airplane is accomplished in its aft section of the propulsion module, with lateral motion constraints only on its forward section.", "\nThe present invention has a disadvantage. ", "It increases the dry weight of the flyback booster. ", "This is a relatively minor matter for a first stage booster because the first stage separates from its payload at about ten percent of orbital energy. ", "If an existing xe2x80x9coff the shelfxe2x80x9d rocket propulsion stage is used as the removable propulsion module, its modification costs will be much less than the development and test costs required to build an integral tank rocket propulsion system. ", "The inventors believe, as a result of their mathematical model of the present invention, that use of a separable rocket propulsion module will add only about ten percent to the dry weight of the flyback booster, compared to its weight if it were to be constructed with integral tanks as is taught by the prior art." ]

{ "pile_set_name": "USPTO Backgrounds" }

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[ "Q:\n\nnewInstance vs new in jdk-9/jdk-8 and jmh\n\nI've seen a lot of threads here that compare and try to answer which is faster: newInstance or new operator.", "\nLooking at the source code, it would seem that newInstance should be much slower, I mean it does so many security checks and uses reflection. ", "And I've decided to measure, first running jdk-8. ", "Here is the code using jmh.", "\n@BenchmarkMode(value = { Mode.", "AverageTime, Mode.", "SingleShotTime })\n@Warmup(iterations = 5, time = 2, timeUnit = TimeUnit.", "SECONDS) \n@Measurement(iterations = 5, time = 2, timeUnit = TimeUnit.", "SECONDS) \n@State(Scope.", "Benchmark) \npublic class TestNewObject {\n public static void main(String[] args) throws RunnerException {\n\n Options opt = new OptionsBuilder().include(TestNewObject.class.getSimpleName()).build();\n new Runner(opt).run();\n }\n\n @Fork(1)\n @Benchmark\n public Something newOperator() {\n return new Something();\n }\n\n @SuppressWarnings(\"deprecation\")\n @Fork(1)\n @Benchmark\n public Something newInstance() throws InstantiationException, IllegalAccessException {\n return Something.class.newInstance();\n }\n\n static class Something {\n\n } \n}\n\nI don't think there are big surprises here (JIT does a lot of optimizations that make this difference not that big):\nBenchmark Mode Cnt Score Error Units\nTestNewObject.newInstance avgt 5 7.762 ± 0.745 ns/op\nTestNewObject.newOperator avgt 5 4.714 ± 1.480 ns/op\nTestNewObject.newInstance ss 5 10666.200 ± 4261.855 ns/op\nTestNewObject.newOperator ss 5 1522.800 ± 2558.524 ns/op\n\nThe difference for the hot code would be around 2x and much worse for single shot time.", "\nNow I switch to jdk-9 (build 157 in case it matters) and run the same code.", "\nAnd the results:\n Benchmark Mode Cnt Score Error Units\n TestNewObject.newInstance avgt 5 314.307 ± 55.054 ns/op\n TestNewObject.newOperator avgt 5 4.602 ± 1.084 ns/op\n TestNewObject.newInstance ss 5 10798.400 ± 5090.458 ns/op\n TestNewObject.newOperator ss 5 3269.800 ± 4545.827 ns/op\n\nThat's a whooping 50x difference in hot code. ", "I'm using latest jmh version (1.19.SNAPSHOT).", "\nAfter adding one more method to the test:\n@Fork(1)\n@Benchmark\npublic Something newInstanceJDK9() throws Exception {\n return Something.class.getDeclaredConstructor().newInstance();\n}\n\nHere are the overall results n jdk-9:\nTestNewObject.newInstance avgt 5 308.342 ± 107.563 ns/op\nTestNewObject.newInstanceJDK9 avgt 5 50.659 ± 7.964 ns/op\nTestNewObject.newOperator avgt 5 4.554 ± 0.616 ns/op \n\nCan someone shed some light on why there is such a big difference?", "\n\nA:\n\nFirst of all, the problem has nothing to do with the module system (directly).", "\nI noticed that even with JDK 9 the first warmup iteration of newInstance was as fast as with JDK 8.", "\n# Fork: 1 of 1\n# Warmup Iteration 1: 10,578 ns/op <-- Fast!", "\n# Warmup Iteration 2: 246,426 ns/op\n# Warmup Iteration 3: 242,347 ns/op\n\nThis means something has broken in JIT compilation.", "\n-XX:+PrintCompilation confirmed that the benchmark was recompiled after the first iteration:\n10,762 ns/op\n# Warmup Iteration 2: 1541 689 ! ", " 3 java.lang.", "Class::newInstance (160 bytes) made not entrant\n 1548 692 % 4 bench.generated.", "NewInstance_newInstance_jmhTest::newInstance_avgt_jmhStub @ 13 (56 bytes)\n 1552 693 4 bench.generated.", "NewInstance_newInstance_jmhTest::newInstance_avgt_jmhStub (56 bytes)\n 1555 662 3 bench.generated.", "NewInstance_newInstance_jmhTest::newInstance_avgt_jmhStub (56 bytes) made not entrant\n248,023 ns/op\n\nThen -XX:+UnlockDiagnosticVMOptions -XX:+PrintInlining pointed to the inlining problem:\n1577 667 % 4 bench.generated.", "NewInstance_newInstance_jmhTest::newInstance_avgt_jmhStub @ 13 (56 bytes)\n @ 17 bench.", "NewInstance::newInstance (6 bytes) inline (hot)\n ! ", " @ 2 java.lang.", "Class::newInstance (160 bytes) already compiled into a big method\n\n\"already compiled into a big method\" message means that the compiler has failed to inline Class.newInstance call because the compiled size of the callee is larger than InlineSmallCode value (which is 2000 by default).", "\nWhen I reran the benchmark with -XX:InlineSmallCode=2500, it became fast again.", "\nBenchmark Mode Cnt Score Error Units\nNewInstance.newInstance avgt 5 8,847 ± 0,080 ns/op\nNewInstance.operatorNew avgt 5 5,042 ± 0,177 ns/op\n\nYou know, JDK 9 now has G1 as the default GC. ", "If I fall back to Parallel GC, the benchmark will also be fast even with the default InlineSmallCode.", "\nRerun JDK 9 benchmark with -XX:+UseParallelGC:\nBenchmark Mode Cnt Score Error Units\nNewInstance.newInstance avgt 5 8,728 ± 0,143 ns/op\nNewInstance.operatorNew avgt 5 4,822 ± 0,096 ns/op\n\nG1 requires to put some barriers whenever an object store happens, that's why the compiled code becomes a bit larger, so that Class.newInstance exceeds the default InlineSmallCode limit. ", "Another reason why compiled Class.newInstance has become larger is that the reflection code had been slightly rewritten in JDK 9.", "\n\nTL;DR JIT has failed to inline Class.newInstance, because InlineSmallCode limit has been exceeded. ", "The compiled version of Class.newInstance has become larger due to changes in reflection code in JDK 9 and because the default GC has been changed to G1.", "\n\nA:\n\nThe implementation of Class.newInstance() is mostly identical, except the following part:\n\nJava 8:\n\nConstructor<T> tmpConstructor = cachedConstructor;\n// Security check (same as in java.lang.reflect.", "Constructor)\nint modifiers = tmpConstructor.getModifiers();\nif (!", "Reflection.quickCheckMemberAccess(this, modifiers)) {\n Class<?", "> caller = Reflection.getCallerClass();\n if (newInstanceCallerCache !", "= caller) {\n Reflection.ensureMemberAccess(caller, this, null, modifiers);\n newInstanceCallerCache = caller;\n }\n}\n\nJava 9\n\nConstructor<T> tmpConstructor = cachedConstructor;\n// Security check (same as in java.lang.reflect.", "Constructor)\nClass<?", "> caller = Reflection.getCallerClass();\nif (newInstanceCallerCache !", "= caller) {\n int modifiers = tmpConstructor.getModifiers();\n Reflection.ensureMemberAccess(caller, this, null, modifiers);\n newInstanceCallerCache = caller;\n}\n\nAs you can see, Java 8 had a quickCheckMemberAccess which allowed to bypass the expensive operations, like Reflection.getCallerClass(). ", "This quick check has been removed, I’d guess, because it wasn’t compatible with the new module access rules.", "\nBut there’s more to it. ", "The JVM might optimize reflective instantiations with a predictable type and Something.class.newInstance() refers to a perfectly predictable type. ", "This optimization might have become less effective. ", "There are several possible reasons:\n\nthe new module access rules complicate the process\nsince Class.newInstance() has been deprecated, some support has been deliberately removed (seems unlikely to me)\ndue to the changed implementation code shown above, HotSpot fails to recognize certain code patterns that trigger the optimizations\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "Optimization of a generalized radial-aortic transfer function using parametric techniques.", "\nThe central aortic blood pressure (cBP) waveform, which is different to that of peripheral locations, is a clinically important parameter for assessing cardiovascular function, however the gold standard for measuring cBP involves invasive catheter-based techniques. ", "The difficulties associated with invasive measurements have given rise to the development of a variety of noninvasive methods. ", "An increasingly applied method for the noninvasive derivation of cBP involves the application of transfer function (TF) techniques to a non-invasively measured radial blood pressure (BP) waveform. ", "The purpose of the current study was to investigate the development of a general parametric model for determination of cBP from tonometrically transduced radial BP waveforms. ", "The study utilized simultaneously measured invasive central aortic and noninvasive radial BP waveform measurements. ", "Data sets were available from 92 subjects, a large cohort for a study of this nature. ", "The output error (OE) model was empirically identified as the most appropriate model structure. ", "A generalized model was developed using a pre-specified derivation cohort and then applied to a validation data set to estimate the recognized features of the cBP waveform. ", "While our results showed that many relevant BP parameters could be derived within acceptable limits, the estimated augmentation index (AI) displayed only a weak correlation compared to the invasively measured value, indicating that any clinical diagnosis or interpretation based on estimated AI should be undertaken with caution." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Tasia Valenza\n\nTasia Valenza (born April 5, 1967 in New York City) is an American actress and voice actress.", "\n\nPersonal life \nTasia Valenza is the daughter of former actor and restaurant consultant Frank Valenza and Gloria Valenza. ", "Her twin brother is a former television writer and now solar marketing consultant Tor Alexander Valenza. ", "She also has an older brother, Greg.", "\n\nCareer \nIn the early part of her career, Valenza played the role of Dottie Thornton on All My Children from 1982 to 1986, earning a Daytime Emmy nomination for her work. ", "She also showed up in a guest star appearance in the fifth season of The A-Team, and a recurring role as Lieutenant Winslow in Space: Above and Beyond. ", "She also portrayed Jodie Abramovitz in Aaron Spelling's drama series The Heights. ", "Valenza is best known to play female villains in video games such as Batman: Arkham Asylum and multiple Spider-Man games.", "\n\nValenza appeared in the 1988 Star Trek: The Next Generation episode \"Coming of Age\" as a Vulcan. ", "She returned to Star Trek in 2017 with a voiceover role in the first two episodes of Star Trek: Discovery as the computer voice of the USS Shenzhou.", "\n\nValenza is married to Harvey Stern since 1995 and has three children.", "\n\nFilmography\n\nFilm\nCrackers – Maria\nRappin' – Dixie\n\nTelevision\n\n21 Jump Street – Lacey King\nAdam-12 – Lucinda Caralis\nThe A-Team – Bonita\nAll My Children – Dottie Thornton Martin \nThe Bold and the Beautiful – Suzanne\nCheers – Customer\nDirty Little Secret – Malina, Admirer\nDanger Rangers – Gabriella, Firefighter #3\nFame – Denise Hudson\nGood Grief – Elke\nThe Heights – Jodie Abramowitz\nHighway to Heaven – Maria Rojas\nHunter – Lupe Ortiz\nMy Demon Lover – Miguela\nOne West Waikiki – Amy Shigeta\nSometimes They Come Back – Kate\nSpace: Above and Beyond – Lt. ", "Kelly Anne Winslow\nSupercarrier – Extra\n\nAnimation\n\nVideo games\n\nAge of Empires III – Amelia Black\nAge of Mythology – Reginleif\nAge of Mythology: The Titans – Reginleif\nArcanum: Of Steamworks and Magick Obscura – Raven\nBattlezone II: Combat Commander – Yelena Shabayev\nDanger Rangers – Gabriella\nThe Elder Scrolls Online – Additional Voices\nThe Evil Within – Myra Hanson\nKing's Quest: Mask of Eternity – Sara, Sylph\nMetal Gear Solid: Integral – Sniper Wolf \nNeverwinter Nights 2: Mask of the Betrayer – Nadaj, Female Hardened Battler, Female Dwarf\nSaints Row 2 – Monica Hughes\nSamurai Western – Claudia, Child #2\nStormrise – Vantage\nToo Human – Hel\n\nReferences\n\nExternal links\nOfficial website\n\n Tazia Valenza at Behind The Voice Actors\n\nCategory:1967 births\nCategory:Actresses from New York City\nCategory:American soap opera actresses\nCategory:American television actresses\nCategory:American child actresses\nCategory:American video game actresses\nCategory:American voice actresses\nCategory:Living people\nCategory:20th-century American actresses\nCategory:21st-century American actresses\nCategory:Hispanic and Latino American actresses\nCategory:American people of Costa Rican descent" ]

{ "pile_set_name": "Wikipedia (en)" }

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[ "Aldeburgh dig unearths teenager's 'keepsakes' box Published duration 7 October 2013\n\nimage caption The dig took place at Barber's Point on the north bank of the River Alde near Aldeburgh\n\nimage caption An unexpected discovery was a young woman's trinkets, including what was initially thought to be a duck egg\n\nimage caption The 'keepsakes' appeared to have been in a wooden box buried in a grave\n\nimage caption Seven skeletons had been found on previous digs, and more have been uncovered\n\nimage caption Once it was fully unearthed, the duck egg turned out to be a cowry shell, which was believed to originate from the Indian Ocean\n\nimage caption About 40 volunteers worked with Suffolk County Council's archaeological services team\n\nAn Anglo-Saxon teenage girl's box of trinkets is thought to have been uncovered by archaeologists during a three-week dig in Suffolk.", "\n\nThe excavation of a graveyard, dating from about AD650, has been completed at Barber's Point on the River Alde.", "\n\nEight more skeletons have been found in graves alongside seven others which were uncovered during previous digs.", "\n\nThe box of 'keepsakes' included a bracelet, a brooch and a cowry shell which is almost completely intact.", "\n\nThe graveyard, near Aldeburgh, is believed to be one of the earliest examples of a Christian, rather than Pagan, burial site in East Anglia.", "\n\nOther graves were found without skeletons, which are believed to have decayed in the acidic soil.", "\n\nimage caption A dolphin ornament found at the latest dig dates from Roman times\n\nJezz Meredith, from Suffolk County Council's Archaeological Service, said: \"The group of items found are likely to be keepsakes or mementos placed at the feet of this young adult female.", "\n\n'Exceeded expectations'\n\n\"[It's] very different from the sorts of things placed with the dead in the earlier Pagan period. \"", "\n\n\"Before the Christian era, males were buried with weapons and females with their finery - so they were equipped and armed for the next world.\"", "\n\nThe service has been working alongside the Aldeburgh & District Local History Society (ADLHS) using Heritage Lottery Fund money.", "\n\nTony Bone, chairman of ADLHS, said: \"We've done four digs here from 2004-2010 and it's great we've come to a conclusion.", "\n\n\"It's exceeded our expectations and we look forward to the deliberations of the county team to tell us more about these finds.\"", "\n\nThe shell was removed from the site encased in earth and the team initially thought it was a duck egg.", "\n\nThe cowry shell was believed to have originated from the Indian Ocean and the team said they had been found in other Saxon graves.", "\n\nThe dig also uncovered a dolphin ornament dating from Roman occupation of the site.", "\n\nThe skeletons and artefacts have been removed by the county team and may eventually end up on public display." ]

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[ "Risk of Sensorineural Hearing Loss and Bilirubin Exchange Transfusion Thresholds.", "\nHigh bilirubin levels are associated with sensorineural hearing loss (SNHL). ", "However, few large studies of relative and excess risk exist. ", "We sought to quantify the risk of SNHL in newborns who had bilirubin levels at or above American Academy of Pediatrics exchange transfusion thresholds (ETT). ", "Infants born at ≥35 weeks gestation in 15 Kaiser Permanente Northern California hospitals from 1995-2011 were eligible (N = 525 409). ", "We used a nested double cohort design. ", "The exposed cohort included subjects with ≥1 bilirubin level at or above ETT. ", "The unexposed cohort was a 3.6% random sample of subjects with all bilirubin levels below ETT (10 unexposed per exposed). ", "An audiologist, blinded to bilirubin levels, reviewed the charts of children in whom SNHL had been diagnosed before age 8 years to confirm the diagnosis. ", "We calculated Cox proportional hazard ratios for time to diagnosis of SNHL. ", "SNHL was confirmed in 11 (0.60%) of the 1834 exposed subjects and in 43 (0.23%) of the 19 004 unexposed. ", "Only bilirubin levels ≥10 mg/dL above ETT were associated with a statistically significant increased risk of SNHL (hazard ratio: 36 [95% confidence interval (CI): 13 to 101]). ", "Likewise, only bilirubin levels ≥35 mg/dL were associated with a statistically significant increased risk of SNHL (hazard ratio: 91 [95% CI: 32 to 255]). ", "For subjects with total serum bilirubin levels 0 to 4.9 mg/dL above ETT, the upper limit of the 95% CI for excess risk was 0.5%. ", "Only bilirubin levels well above ETT were associated with SNHL. ", "At lower bilirubin levels, the excess risk of SNHL was low." ]

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[ "The trailer provided a detailed look at Mario’s latest adventure, and it looks completely and utterly insane.", "\n\nMy first impression was one of confusion: why the hell was Mario running around New York City? ", "Are those supposed to be real people? ", "Why the ‘effin heck is he jumping on taxi cabs? ", "WHAT THE HELL IS THIS!?", "\n\nSafe to say that everyone else watching the trailer was thinking something along those lines, curious as to what they were actually witnessing. ", "It was equal parts weird as it was intriguing, and it turns out that the concept for the title is just as crazy as the very thought of Mario walking around a real city is.", "\n\nSuper Mario Odyssey has been labelled as “A new Mario game” by Nintendo, which I gather means more than it simply being, you know, a new Mario game. ", "I think Nintendo means this is a new type of Mario game. ", "All it takes is a glance at how weird Mario looks standing next to realistically proportioned humans to understand that, yes, this is indeed a new type of Mario game.", "\n\nPerhaps most striking about Odyssey is the sheer diversity of the landscapes. ", "Mario moves from New York City to a whole host of different and crazy worlds, each more different than the next. ", "Unlike Super Mario Sunshine and Super Mario Galaxy, Odyssey promises to take Mario to not just one unfamiliar world, but multiple.", "\n\nNintendo says that Mario’s hat, which strangely has eyes, is sentient and can do things “only possible on Switch”, whatever that means. ", "The trailer shows Mario using the hat in a number of different ways, including jumping off it in midair, and throwing it like a boomerang.", "\n\nIf you haven’t yet seen the trailer, check it out below. ", "Prepared to be blown away. ", "It unfortunately won’t be a launch title, and instead will hit Switch in late 2017.", "\n\nGaetano Prestia Editor in Chief\n\nGaetano loves Doritos and always orders Mountain Dew with his KFC. ", "He's not sorry. ", "He also likes Call Of Duty, but would much rather play Civ. ", "He hates losing at FIFA, and his pet hate is people who recline their seat on short-haul flights." ]

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[ "Chess (poem)\n\nChess () is a poem written by Jan Kochanowski, first published in 1564 or 1565. ", "Inspired by Marco Girolamo Vida's Scacchia Ludus, it is a narrative poetry work that describes a game of chess between two men, Fiedor and Borzuj, who fight for the right to marry Anna, princess of Denmark. ", "The poem anthropomorphises the pieces, presenting the game as a battle between two armies, in a style reminiscent of battle scenes in the works of Homer and Virgil.", "\n\nIn 1912, Alexander Wagner reconstructed the game described in the poem, while Yuri Averbakh found that it has three possible endings in 1967.", "\n\nBackground\nDuring his visit to the Italian lands in 1558–59, Jan Kochanowski encountered Scacchia Ludus, a poem by Marco Girolamo Vida, which describes a game of chess played on Mount Olympus between Apollo and Mercury. ", "It is possible that Kochanowski also met Vida himself. ", "Inspired by Scacchia Ludus, Kochanowski decided to create his own poem with a chess game as the main topic.", "\n\nUntil the second half of the 19th century, Chess was often thought to be only a paraphrase of Vida's poem, rather than an independent work. ", "In his 1856 essay Chess in Poland (), historian Maurycy Dzieduszycki proved that Kochanowski's poem, while inspired by Scacchia Ludus, is dependent on it only to a small extent and should be considered a fully separate work of art.", "\n\nChess is dedicated to \"Jan Krzysztof, count of Tarnów, castellan of Wojnicz.\"", "\n\nPlot\n\nTarses, the king of Denmark, has a daughter named Anna. ", "Of the many foreigners who wanted to marry her, two men, Fiedor and Borzuj, stood out as the most interested. ", "After the two men proclaim that they want to have a duel, Tarses proposes a game of chess instead. ", "Fiedor and Borzuj learn the rules and after spending some time training, meet in Tarses' palace for the game.", "\n\nBorzuj is chosen to play white and Fiedor black. ", "The game, described as a battle of anthropomorphised pieces starts, comprising the biggest part of the poem. ", "Kochanowski uses the names of the pieces (which are often different from their equivalents in modern Polish) in literal sense, describing the game as a war between infantry soldiers (pawns), knights, bow-wielding priests (bishops), war elephants (rooks), queens and kings. ", "The play is described metaphorically: Captures are referred to as killings, the pieces are said to have emotions. ", "At one point, Borzuj attempts to \"resurrect\" a dead knight, i.e. illegally put a captured piece on the board. ", "When both queens are killed, the kings are said to look for new wives among their servants – a metaphor of pawn promotion.", "\n\nThe play advances into the endgame and Borzuj indeed promotes and gets a new queen. ", "The match progresses to a position which seems to end with white inevitably mating in its next move (see the diagram). ", "Fiedor sees no way to evade his loss and makes no move, despite urging comments from Borzuj and advice to resign from members of Tarses' court who watch the game. ", "The sun sets with black still not making any move and it is decided that the players will stop the game for the night, to resume on the following day.", "\n\nLater Anna, who would rather marry Fiedor, visits the room with the chessboard, guarded for the night. ", "She also initially believes that there is no way to evade white's mate, but subsequently notices a chance for black. ", "Anna loudly expresses an enigmatic opinion: That knights know how to fight, priests are good at giving advice, infantry doesn't hesitate to walk forward and that it's no loss to change a dear thing for someone beloved. ", "Before leaving, the princess turns the black rook to the side.", "\n\nThe following day, hopeless Fiedor and content Borzuj meet to finish their game. ", "Fiedor asks the guards about the turned rook and learns about Anna's visit and her words. ", "Borzuj pays no attention, believing she was referring to the real-life knights and priests. ", "Fiedor starts to think deeply and understands that the princess' words are a message with a hidden hint for him: that he should not use the knight and the bishop, but sacrifice the most valuable piece his still has, the rook; before ending the game using the pawns. ", "To get the attention of his spectators, Fiedor proclaims that he will admit loss if he won't be able to mate his opponent in the next three moves. ", "To Borzuj's surprise, black sacrifices the rook and mates with two pawns. ", "Fiedor marries Anna, while Borzuj leaves, declining an invitation to the wedding.", "\n\nCritical analysis\n\nArtistic\nChess is written in eleven-syllable verse. ", "It was one of the first Polish language works created by Kochanowski, who was earlier mostly writing elegies and epigrams in Latin. ", "According to prof. ", "Edmund Kotarski of the Gdańsk University, Chess resembles a short story in the parts dealing with the human characters, while the battle of the chess pieces is a parody of heroic epos of Homer and Virgil, \"following its style while presenting a plot which clearly was not monumental or grand\", so that the \"clash between seriousness and humour\" results in humorous effects. ", " The poem was described as more humane that Scacchia Ludus which inspired it, as instead of mythological gods, Kochanowski tells a story of two young men fighting for their future.", "\n\nChess\nChess theoretician Władysław Litmanowicz considers Borzuj and Fiedor's playing skills as not extraordinary, but notices that the game's theory was much less advanced in Kochanowski's times compared to the modern day.", "\n\nThe ending combination is a variation of the well-known Dilaram problem. ", "Anna's advice was most likely easily understood by the contemporary readers.", "\n\nReconstruction\n\nBackground\nIn 1912, the Szachista Polski () magazine announced a contest to reconstruct the game described in Kochanowski's poem. ", "It ended with only one entry sent – created by Alexander Wagner, who was one of the magazine's employees. ", " Wagner submitted his reconstruction under the pseudonym of \"Wanda Reger Nelska\" (an anagram of his name), so that his colleagues wouldn't be biased in judging the entries. ", "Most of the commentary in Wagner's work are simply quotations from the poem.", "\n\nWagner's reconstruction\n\n1. ", "d4 d5 2. ", "c3 e5 3. ", "e3 a5 4. ", "b3 h6 5. ", "a3 Na6 6. ", "h3 Bf5 7. ", "Be2 Qd6 8. ", "c4 g6 9. ", "b4 axb4 10. ", "axb4 O-O-O 11. ", "Nf3 Kb8 12. ", "Nc3 Rh7 13. ", "Nxd5 Nxb4 14. ", "Nxe5 Nc2+ 15. ", "Kf1 Nxa1 16. ", "Bd2 g5 17. ", "Qxa1 Be6 18. ", "e4 Bg7 19. ", "Ne3 f6 20. ", "Nf3 Qa6 21. ", "Qb2 Qd6 22. ", "d5 Bd7 23. ", "Qd4 Qb6 24. ", "c5 Qb1+ 25. ", "Be1 Be8 26. ", "c6 Rc8 27. ", "d6 f5\n\nBorzuj touches his e4 pawn, thinking about 28. ", "exf5?. ", "Fiedor immediately captures the white queen with 28. ... ", "Bxd4. ", "Borzuj protests, as he is not aware of the touch-move rule. ", "They eventually agree to cancel the move.", "\n\n28. ", "Qc4 Nf6 29. ", "Ne5 Qb6 30. ", "Rh2 h5 31. ", "f3 Rd8 32. ", "Bg3 f4\n\nWhite contemplates whether he would rather want to lose a knight or a bishop and decides to keep the former.", "\n\n33. ", "Nf5 fxg3 34. ", "Nxg3 Rh8 35. ", "Nd7+ Bxd7 36. ", "cxd7 Rxd7 37. ", "Qc1 Qxd6 38. ", "Qxg5 h4 39. ", "Nf5 Qb6 40. ", "Qf4 Rh5 41. ", "Qc1 c5 42. ", "Ne3 Bh6 43. ", "Kg1 Qd8 44. ", "Kf1 Rd2 45. ", "Kf2 Nd5 46. ", "Qc4 Rxe2+ 47. ", "Qxe2 Nxe3 48. ", "Qe1 Qg5 49. ", "Kg1 c4 50. ", "Qf2 c3 51. ", "Rh1 b6 \n\nBorzuj attempts to illegally put his captured knight again on the board, which Wagner renders as 52. ", "Na1–b3?", "\n\n52. ", "Kh2 Qg3+ 53. ", "Qxg3+ hxg3+ 54. ", "Kxg3 Ra5 55. ", "Kf2 Ra8 56. ", "Ke2 Nc2 57. ", "Kd1 Ne3+ 58. ", "Kc1 Nxg2+ 59. ", "Kb1 b5 60. ", "h4 b4 61. ", "h5 b3 62. ", "e5 Bg5 63. ", "h6 Nh4 64. ", "e6 Bf6 65. ", "h7 Nxf3 66. ", "e7 Bxe7 67. ", "h8=Q+ Kb7 68. ", "Qh3 Ne5 69. ", "Qh5 Nd3 70. ", "Qh3 Nb4 71. ", "Qf3+ Kb8 72. ", "Qe2 Bf6 73. ", "Qe6 Bg7 74. ", "Qd7 Bd4 75. ", "Rh7\n\nThe play is suspended for the night after this move.", "\n\n75. ... ", "Ra1+ 76. ", "Kxa1 b2+ 77. ", "Kb1 c2#\n\nAlternative endings\nIn his book В поисках истины (), Grandmaster Yuri Averbakh also reconstructed the game. ", "At the point when the play is stopped for the night, it is a \"mirror reflection\" of Wagner's work (with the black king on g8, black rook on h8 etc.). ", "However, Averbakh proposed three alternative ways for black to mate. ", "The first one is the same as Wagner's.", "\n\nThe two new variants are (starting from Wagner's reconstruction after 75. ", "Rh7):\n\n75. ... ", "c2+ 76. ", "Kc1 Ra1+ 77. ", "Kd2 c1=Q+ 78. ", "Ke2 Qd1# (or Qd3#) and\n\n75. ... ", "Ra1+ 76. ", "Kxa1 c2+ 77 Qxd4 c1=Q#\n\nNote that the first variant would exceed Fiedor's claim that he'll mate within three moves.", "\n\nSee also\n Chess in the arts and literature\n\nReferences\n\nFurther reading\n \n With foreword by Julian Krzyżanowski.", "\n\nExternal links\n\n Review of Kochanowski's works, including Chess\n\nCategory:1565 books\nCategory:16th century in chess\nCategory:Chess in Poland\nCategory:Polish poems\nCategory:Poems about chess" ]

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[ "Official Web Site for Bennche the COWBOY Side by Side Utility Vehicles: UTVs by Bennche. ", "The Affordable Working SXS and SPIRE Sport Style Utility Side by Side Vehicles. ", "The Mini UTVs. - ", "lineup of COWBOY, SPIRE, BIGHORN, GRAY WOLF ATV." ]

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[ "Novotel Salerno Est Arechi\n\nSalerno, Italia\n\nThe Novotel Salerno is a 4-star hotel with sea views, a short distance from Arechi stadium. ", "Easily reached from the Salerno SS18 ring road, Mariconda exit, it is just 3. ", "7 miles (6 km) from the city center. ", "The hotel has 116 air-conditioned rooms, 1 restaurant, 1 bar and 5 conference rooms for meetings or conferences (seating for 180). ", "Ideal for business trips or for couples or families on vacation. ", "Novotelcafé with 24-hour catering. ", "Outdoor pool (open May to September), fitness center, hammam and car park\n\nThis stylish hotel has an ideal location in Salerno , close to the Amalfi Coast and the archaeological marvels of Paestum and Pompeii. ", "And after so much sightseeing, you'll deserve to relax in our spa area! ", "The novotelcafé offers the flavors and colors of the South." ]

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[ "Victoria is the ninth-best city in Canada for youth to work according to a report from Youthful Cities. ", "It lands in the middle of a list of 21 cities total. (", "Shalu Mehta/News Staff)\n\nGood news, youths.", "\n\nVictoria ranked in the top 10 cities for people age 15 to 29 to work in Canada.", "\n\nYouthful Cities, an organization aiming to lead urban regeneration towards youthfulness, ranked Victoria as the ninth-best city for young Canadians to work in its 2019 Urban Work Index survey.", "\n\nOut of a possible 1,310 points that help evaluate affordability, education, employment and entrepreneurship, Victoria scored around 635.", "\n\nAccording to the survey which looked at 21 different cities, Victoria has the second-lowest average student debt and the seventh-lowest tuition cost. ", "It also has the best performance in year-over-year change in youth full-time jobs as a per cent of total jobs.", "\n\nREAD MORE: Victoria ranked third-best city in Canada to be a woman\n\nThe survey says: “Victoria excels as a place for youth to navigate the education to work transition.” ", "It also boasts affordable education, better access to education and good work-integrated learning so youths can develop work-ready skills while studying.", "\n\nSome areas that need improvement in Victoria are related to the cost of food, clothing and housing, placing Victoria in the middle of the 21 cities on the list.", "\n\nThe organization defines youths as 15 to 29-year-olds and says youths make up about 18 per cent of Victoria’s population.", "\n\nIn a statement from Youthful Cities co-founder Rober Barnard, he says Canadian youths will be the most educated ever but will also be the most indebted youth ever. ", "He also says the youth unemployment rate is double the national average in Canada.", "\n\n“Youth clearly want more affordable housing, less student debt, good jobs and cities that really support entrepreneurs,” Barnard said.", "\n\nREAD MORE: Canada sees second straight monthly employment surge with 55,900 net new jobs\n\nEdmonton was ranked as the best city for urban work for youth, followed by Montreal and Ottawa.", "\n\nToronto came in one notch below Victoria at number 10 and Saskatoon, Halifax and Winnipeg made up the bottom three on the list.", "\n\nEdmonton scored high when it came to affordability for food, housing, health, utilities, education and transportation. ", "The survey also said Edmonton is a “great option for youth looking for an entrepreneurial environment.”", "\n\nOverall, every region of Canada has at least one city on the list. ", "Each city that was analyzed was chosen based on population, geographical representation within a region and local engagement.", "\n\nThe full Youthful Cities report can be found here.", "\n\nshalu.mehta@goldstreamgazette.com\n\nLike us on Facebook and follow us on Twitter" ]

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[ "Q:\n\njQuery $.data() not returning correct value when data attribute is updated\n\nI have a very strange issue. ", "Im not sure why the $.data not returning the correct value after updating the data attibute even though the element has the correct value in firebug.", "\nDemo: http://jsfiddle.net/gv5cR/\n<div>\n <input type=\"text\" name=\"price\" id=\"price\"/>\n</div>\n<button type=\"button\" id=\"submit\" data-price=\"100\">Submit</button>\n<div id=\"result\"></div>\n<script>\n$(document).ready(function(){\n $('#price').change(function(){\n\n $('#submit').attr('data-price',$(this).val());\n\n });\n\n $('#submit').click(function(){\n $('#result').html($(this).data('price'));\n });\n\n});\n<script>\n\nA:\n\nYou should set data using .data()\n$('#submit').data('price',$(this).val());\n\nDemo ---> http://jsfiddle.net/gv5cR/2/\n\n" ]

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[ "Brandon police have cracked a mysterious case and now a woman they needed help identifying is facing mischief charges for claiming to be a human trafficking victim when she was actually on the run from a custody battle in Ontario.", "\n\nThe woman in the photo has been identified as a 39-year-old mother from Etobicoke. ", "She is now charged with public mischief for allegedly lying to investigators about being the victim of human trafficking. ", "She claimed she was abducted as a child and had been held captive in Winnipeg for years.", "\n\n'She just had to keep up with her story and she was fairly good at it.' - ", "Sgt. ", "Kevin McLean , Brandon police\n\n\"This female came to Brandon and made allegations regarding human trafficking involving herself and her daughter,\" said Brandon police Sgt. ", "Kevin McLean, adding the case stumped investigators since September.", "\n\n\"These allegations have been proven false and in fact she is fleeing Ontario with her daughter and it's something to do with a custody order out of Ontario.\"", "\n\nWoman 'good at' telling false story\n\nThe woman in this photo, sent out by Brandon police when they were trying to identify her, is now charged with public mischief for allegedly lying to investigators about being the victim of human trafficking. (", "Courtesy Brandon Police Service) The woman and her daughter came to Brandon in late August with no identification. ", "They reached out to a social agency in town. ", "That agency in turn called police.", "\n\n\"I believe she started this story with the service agency that took her in when she first got to Brandon and then they got in contact with us because it was criminal in nature. ", "So at that point, she just had to keep up with her story and she was fairly good at it,\" McLean said.", "\n\n\"Her story didn't involve names. ", "She didn't know names,\" said McLean. \"", "So before we could start an investigation, we had to find out who they were.\"", "\n\nFirst, investigators reached out to police agencies across Canada to try and match their alleged victim to any of the child abductions they might have had on file but things weren't coming together.", "\n\nInvestigators had suspicions\n\n\"Nobody had anything, so at that point we have to expand things to find out who these people are,\" said McLean.", "\n\nHe added the woman kept to her story and gave investigators a false name.", "\n\n\"Our investigators had suspicions that possibly it wasn't a true story but we have to go with every, with somebody's story until we prove otherwise,\" said McLean.", "\n\nPolice got a call from the Canadian Centre for Child Protection just before 6 p.m. Thursday notifying them a social worker from Ontario, who had previous contact with the mother and daughter, had come forward.", "\n\nThat information eventually led to the woman's arrest.", "\n\n'Quite shocking news'\n\n\"I know she was charged with mischief. ", "I don't know if more charges will follow,\" an Ontario man who says he is the woman's ex and the girl's dad told CBC News.", "\n\nThe man called the revelations \"quite shocking news.\"", "\n\nThe two broke up in 2008 and reconciled before breaking up again in 2009, he said, adding he'd had unsupervised visits with his daughter since 2010.", "\n\nThe only victim in this entire incident is that young seven year old. - ", "Sgt. ", "Kevin McLean\n\n\"I had her every other weekend plus 3½ weeks vacation,\" he said, adding that all changed in July after the child made allegations against him.", "\n\n\"My daughter was making allegations that I touched her [inappropriately] and she was saying this herself. ", "This was after being with [her mom] for three weeks during their vacation time,\" he said.", "\n\nA judge at one point granted the woman a restraining order against her ex. ", "Children's Aid Society of Toronto launched an investigation but found the allegations \"unverified,\" he said. ", "His weekly access to the girl resumed Sept 2, he said.", "\n\n\"That's when they disappeared,\" he said.", "\n\nIn the two months the pair were in Manitoba, they were never reported missing. ", "Toronto police are investigating but no charges have been laid. ", "Officers couldn't say if they were considering any charges.", "\n\nThe man said his daughter will be returned to Ontario next week and will be placed in foster care until custody is sorted out.", "\n\n\"She had custody but that will be changed,\" he said. \"", "CAS will be having temporary custody for a month or two as they transition into having my daughter reunited with my family, at which point I will have custody.\"", "\n\nChild 'only victim' here, police say\n\n\"The only victim in this entire incident is that young seven year old,\" said McLean.", "\n\n\"She was dragged along in it, she was coached what to tell people.\"", "\n\nBecause the child is now in CFS custody, the pair can't be identified." ]

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[ "C’est Arnaud Montebourg qui s’en prend à Vincent Bolloré, sans le nommer, lors du deuxième débat de la primaire de la gauche, en rappelant que l’actionnaire d'iTélé \"est allé jusqu'à détruire son outil de travail pour empêcher le pluralisme\", référence aux 31 jours de grève et au départ de près de 100 journalistes. ", "C’est Marine Le Pen qui se prend de bec avec Jean-Jacques Bourdin. \"", "Vous croyez qu’avec mon portable, je reçois des ordres de Monsieur Drahi ? ", "demande le journaliste. ", "Franchement ?\" - \"", "Oui, je crois que les gens qui sont propriétaires des journaux et des radios ont une influence sur la la ligne éditoriale\". ", "C’est Benoît Hamon qui veut imposer une loi anti-concentration car \"ce contre-pouvoir des médias – fondamental dans toute démocratie – suppose que la qualité de l’information ne soit l’objet d’aucun soupçon\". ", "Bref, la question de l’actionnariat des médias s’est invitée dans la campagne.", "\n\nEn très peu de temps, le paysage a subi une mutation technologique et capitalistique. ", "Aujourd’hui, la quasi totalité des médias se répartit entre huit milliardaires dont le point commun est d’avoir bien d’autres activités : BTP, télécoms, luxe … Certains sont là par héritage comme Arnaud Lagardère ou Martin Bouygues; d’autres, par choix comme Xavier Niel (coactionnaire avec Matthieu Pigasse et Pierre Bergé de \"L’Obs\") ou Patrick Drahi. ", "Qui sont-ils ? ", "Pourquoi ont-ils jeté leur dévolu sur ces outils d’influence ? ", "Comment leurs rédactions traitent-elles des sujets qui les fâchent ? ", "De l’actualité de leurs concurrents ?", "\n\nAmaury de Rochegonde et Richard Sénéjoux, respectivement journalistes à \"Stratégies\" et à \"Télérama\" livrent une enquête intitulée \"Médias, les nouveaux empires\". ", "Bonnes feuilles.", "\n\n(ROMUALD MEIGNEUX/SIPA)\n\n\n\n\n\nSerge Dassault\n\nSerge Dassault est propriétaire du Groupe Le Figaro\n\nQuelle relation entretient l’avionneur-patron de presse avec le pouvoir socialiste ? (…) ", "certaines personnalités des gouvernements de François Hollande ont droit à plus d’égard que d’autres, voire à un soutien manifeste. ", "Ce 10 septembre 2012, l’ancien maire d’Évry, Manuel Valls, fraîchement nommé ministre de l’Intérieur, inaugure par exemple la 64e foire de Corbeil-Essonnes en compagnie de Serge Dassault, ancien édile de la ville et parlementaire UMP de l’Essonne. \"", "Je veux dire un mot à Manuel, lance alors au micro le vieux sénateur. ", "Pour moi, pour vous, la sécurité n’est ni de gauche ni de droite. ", "Et je dois vous dire que nous sommes très heureux de son action. ", "C’est pour ça qu’il a l’appui d’un journal bien connu. ", "Mais s’il fait des bêtises, on en reparle. ", "Actuellement, c’est très bien. ", "Pour les Roms et tous les autres, c’est formidable. ", "Donc bravo Manuel, continue !\" ", "Le ministre lève les yeux au ciel, rit de façon un peu gênée et se cache le visage dans les mains. ", "Le voilà prévenu : comme Jean-Yves Le Drian, ministre de la Défense, Manuel Valls ne sera pas la cible des principales attaques du Figaro. ", "Mais il ne doit pas faire de \"bêtises\".", "\n\nMichel Lucas\n\nMichel Lucas, ancien patron du Groupe Crédit Mutuel-CIC, a créé un pôle de presse régionale : \"Les dernières nouvelles d'Alsace\", \"L’Est républicain\", \"Le Progrès\", \"Le Dauphiné libéré\"…\n\nMichel Lucas sait aussi jouer du rapport de forces quand une information extérieure lui déplaît. ", "Le 6 mai 2009, le quotidien \"Les Échos\", qui a osé écrire qu’il était en 2008 \"le dirigeant de banque le mieux payé de France\", avec 1,37 million d’euros de rémunération, l’a appris à ses dépens. ", "Malgré un rectificatif contesté en interne, et publié le lendemain, qui dénonçait l’intégration de sa part variable dans son salaire fixe, le patron décide de supprimer les 1 200 abonnements de son entreprise au quotidien économique et de retirer toutes les publicités du Crédit mutuel-CIC dans le groupe de presse de Bernard Arnault.", "\n\n\n\nMartin Bouygues\n\nMartin Bouygues est actionnaire du groupe TF1 (TF1, TMC, HD1...)\n\nNi Bouygues ni son groupe audiovisuel n’auront à se plaindre du mandat de François Hollande. ", "Au contraire. ", "Sur l’ensemble du quinquennat, on peut même parler d’une certaine bienveillance des gouvernements Ayrault et Valls. ", "Outre l’appui à l’offre de Bouygues sur SFR en 2014 (…) l’entrée (manquée) de Bouygues au capital d’Orange n’a pu être envisagée qu’avec la bénédiction du gouvernement et de l’Élysée. ", "Jusqu’à sa volte-face de fin 2016, François Hollande souhaitait notamment s’appuyer sur cet axe TF1-LCI en vue de la présidentielle. ", "Manuel Valls, sur ce plan, se posait en héritier.", "\n\n\n\n\n\n\n\n(ROMUALD MEIGNEUX/SIPA)\n\nVincent Bolloré\n\nVincent Bolloré est actionnaire de Canal+, C8, CNEWS, CNEWS Matin et d'Havas\n\nHavas se révèle une arme à double tranchant. ", "Le premier groupe publicitaire en France sert aussi, à l’occasion, à marquer son mécontentement envers telle ou telle puissance médiatique. ", "C’est ce que nous a confié Gilles Van Kote, l’ancien directeur du Monde, qui affirme avoir fait face à un véritable \"blocus\" publicitaire de Havas pendant plusieurs mois.", "\n\nEn 2015, selon des données de l’organisme spécialisé Kantar, qui recense les annonces de publicité, les investissements publicitaires de Havas dans le groupe Le Monde ont en effet reculé de 14 %, soit 6 à 7 millions d’euros en net. ", "La raison : un article d’une journaliste pigiste sur le port d’Abidjan, qui aurait fortement déplu à Vincent Bolloré. ", "Un \"papier normal\" selon l’ancien directeur du journal, mais qui aurait eu le don d’énerver sensiblement l’homme d’affaires.", "\n\nBernard Arnault\n\nBernard Arnault est actionnaire des \"Echos\", du \"Parisien-Aujourd'hui en France\", Radio Classique ...\n\nTristan Waleckx, reporter pour Complément d’enquête sur France 2, a fait l’expérience de la capacité du leader mondial du luxe à se protéger contre toute intrusion des journalistes dans son empire. ", "En 2014, (…) (il) remonte ainsi jusqu’à la fabrication des sacs Vuitton en Roumanie et ses nombreux sous-traitants, qui lui permettent d’échapper à la législation française sur le travail, plus contraignante.", "\n\nMais voilà qu’à peine arrivé en Roumanie, le responsable de l’émission Benoît Duquesne (aujourd’hui décédé), l’appelle. ", "Ce dernier a reçu un mail de Nicolas Bazire, le numéro deux de LVMH : \"Je suis très étonné que votre journaliste soit en train de distribuer des liasses de billets pour acheter des témoignages\", lit-il. ", "Quelques jours plus tard, alors que Tristan Waleckx est encore en salle de montage, le même Duquesne revient vers lui : \"Que faisais-tu le 29 janvier dernier ? ", "Je viens d’avoir Arnault et Bazire, qui disent que ce jour-là, tu aurais essayé d’extorquer de l’argent auprès d’un détective privé et qu’ils ont des preuves.\"", "\n\nPar chance, le journaliste constate dans son agenda que ce 29 janvier, il était en tournage à l’Institut d’études politiques de Paris. ", "Et pas à Bordeaux en train de recevoir de l’argent comme le prétendent ses accusateurs. ", "LVMH va jusqu’à porter plainte pour tentative d’extorsion de fonds afin d’empêcher la diffusion du reportage, avant de se désister.", "\n\n\n\n\n\nXavier Niel\n\nXavier Niel est co-actionnaire du groupe \"Le Monde\", \"L'Obs\"...\n\nVolontiers qualifié par la presse de patron \"cool\", d’\"entrepreneur visionnaire\" \"rebelle\" voire \"libertaire\", Xavier Niel se présente ainsi sous un tout autre jour au plus fort de ses batailles.", "\n\nEn janvier 2013, la publication d’une série de six articles intitulés \"Les secrets bien gardés de Xavier Niel\" vaut aussi à Edwy Plenel, directeur de la publication, de découvrir une facette du personnage : \"Dieu sait si nous avons sorti des enquêtes à Mediapart. ", "Nous avons eu 160 procédures judiciaires, et nous n’en avons perdu que trois. ", "Le seul épisode de pression violente que j’ai eu, c’est notre enquête sur Xavier Niel. ", "La seule personne qui m’a assailli de questions au téléphone, qui m’a dit \"je vous enregistre\", c’est lui\"\n\nPensait-il pouvoir bénéficier d’un traitement de faveur compte tenu d’un apport initial de 200.000 euros dans le site en 2008 ?", "\n\n(CHAMUSSY/SIPA)\n\nPatric Drahi\n\nPatrick Drahi est actionnaire de \"Libération\", de \"L'Express\", BFMTV, RMC....\n\n\"Mon aventure dans la presse, elle est totalement anecdotique, expliquait-il le 27 mai 2015 lors d’une audition devant la commission des affaires économiques de l’Assemblée nationale. ", "Je vous raconte l’anecdote parce qu’elle est drôle. ", "Lors d’un entretien avec une journaliste de “Libération”, la demoiselle me dit : “Mais monsieur Drahi, vous allez dépenser 14 milliards pour acheter SFR, nous chez 'Libération', on n’a besoin que de 14 millions pour nous sauver.” ", "ça m’a laissé perplexe, je dois vous l’avouer. ", "Et à l’issue de l’entretien, j’ai dit “prenons le dossier”, parce que si nous pouvions effectivement sauver un titre sans aucune conviction ni médiatique ni politique, pour un millième de l’argent investi dans SFR, je pense qu’on faisait quelque chose de bien.\"", "\n\nBelle histoire. ", "Mais insuffisante pour expliquer l’incursion du milliardaire dans la presse. ", "La réalité est plus prosaïque. ", "C’est plutôt à la faveur d’une rencontre avec François Hollande pour parler câble et télécoms que tout s’est joué. ", "Patrick Drahi a vu que le président se souciait de l’avenir du quotidien en grande difficulté financière. ", "Sa capacité de réaction a fait le reste : \"il a compris qu’on apprécierait [son aide]\", raconte un observateur. ", "Lui a alors apprécié qu’on lui laisse les coudées franches pour la reprise de SFR, les possibles enquêtes fiscales ne devenant alors qu’un lointain souvenir. ", "Emmanuel Macron a alors rejoint le ministre de l’Économie et des Finances, à Bercy, où une enquête a été ouverte par son prédécesseur sur la résidence fiscale de Patrick Drahi en mai 2014.", "\n\nNote des auteurs :\n\nSalariés dans les rédactions de Stratégies (SFR Media) et de Télérama (groupe Le Monde), Amaury de Rochegonde et Richard Sénéjoux tiennent à préciser qu'ils n'ont pas travaillé sur leur groupe respectif afin d'éviter tout conflit d'intérêt. ", "Les pages concernant Xavier Niel et le groupe Le Monde ont été écrites par Amaury de Rochegonde ; celles concernant Patrick Drahi et SFR Media l'ont été par Richard Sénéjoux.", "\n\n\"Medias : Les nouveaux empires\" d'Amaury de Rochegonde et Richard Sénéjoux. ", "First Editions, 2017, 288 p. 16, 95 euros. ", "A paraître le 13 avril." ]

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[ "There goes Viktor Orban again. ", "On October 23, 2018, he used the anniversary of the 1956 Budapest uprisings to compare the EU with the Soviet Union. ", "In so doing, he inferred that both are imperialist occupiers.", "\n\nBy establishing an analogy between the EU and the USSR, Orban makes a mockery of Hungary’s 20th century history. ", "But that apparently does not matter to Orban. ", "Through cheap-shot speechifying, he is trying to set himself up as a defender of European countries against … Europe.", "\n\nWhat’s getting lost in the Orban equation is the vital difference between totalitarianism and democracy. ", "Countries become members of the EU because they joined through a democratic process, the aquis, also meaning they agreed to democratic principles known as the Copenhagen criteria, not because they were colonized.", "\n\nOrban’s revisionist history is straining to fit the EU into a scheme with the USSR. ", "To date, Hungary is not leaving the EU (yet), but Orban is sowing discord for the sole purpose of his personal political advantage at home (and, somewhat ironically, to Russia’s advantage).", "\n\nViktor Orban apparently hates the supranational government, which 83% of the Hungarian population supported joining in the interim—in 2004, when he was not Prime Minister, a post he served from 1998-2002, and 2010-to the present.", "\n\nHe just wants the EU to pay for projects in Hungary, the in-country execution of which he then corruptly funnels to his cronies. ", "But other than taking the money, he wants to have absolute power — no common principles, checks or balances.", "\n\nOrban isn’t a defender of anybody’s rights. ", "He is an unvarnished unilateralist. ", "The European countries and ethnic majorities within those nationalist borders are not enslaved. ", "If anyone, it is Orban who sidelines ethnic and political minorities to his authoritarian control.", "\n\nConclusion\n\nIn short, Orban wants all the benefits and none of the responsibilities of EU membership. ", "That makes Orban a crass opportunist.", "\n\nWhile Orban complains that smaller nations like his Hungary don’t get proper respect inside the EU, consider the case of the Irish: They have gone through hard times, but are happy to be part of the EU. ", "As a tiny island in a globalized world, and historically on the bad end of colonialism from their English neighbors, they know about the benefits of being part of a bigger entity.", "\n\nMr. Orban, meanwhile, sidles up to Vladimir Putin, his fellow world-class cynic. ", "Both are united in one fundamental belief: Let’s get away with whatever we can get away with." ]

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[ "An electronic brake is a device that measures an electrical angle and a current supplied to a motor, generates torque by operating the motor by controlling a current flowing to the motor in accordance with the measured electrical angle, converts the torque into brake pressure through a gear, and brakes a vehicle using the converted brake pressure.", "\nRecently, laws and regulations regarding a brake have been revised to be more severe due to an increase of vehicle accidents, so it is required to increase the pressure of a brake for stopping a vehicle, which was 50 bar, to 90 bar in order to satisfy the revised laws and regulations regarding a brake." ]

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[ "Q:\n\n.Net windows services: install from referenced assembly\n\nThere are lots of examples how to install windows service in one line:\n ManagedInstallClass.", "InstallHelper(\n new[] { Assembly.", "GetExecutingAssembly().Location });\n\nThat works fine until service class is declared in exe module. ", "\nBut the same code doesn't work for me if service class is in referrenced assembly (not declared in executable, but in linked dll). ", "\nIn such a case service is registered as well but can't be started as it is registered with dll path and points to dll (\"service is not a win32 executable\" message appears in event log when I try to start that) \nIf I change GetExecutingAssembly().Location to executable path, then no installers are found and service is not registered at all. ", "\nIs it possible to put service class into referenced assembly and still have ability to register service with minimum effort?", "\nThank you in advance! ", "\n\nA:\n\nhere is some C# code that allows you to install/uninstall a service\" manually\" (without the need to declare custom RunInstaller attributes):\nstatic void InstallService(string path, string name, string displayName, string description)\n{\n ServiceInstaller si = new ServiceInstaller();\n ServiceProcessInstaller spi = new ServiceProcessInstaller();\n si.", "Parent = spi;\n si.", "DisplayName = displayName;\n si.", "Description = description;\n si.", "ServiceName = name;\n si.", "StartType = ServiceStartMode.", "Manual;\n\n // update this if you want a different log\n si.", "Context = new InstallContext(\"install.log\", null);\n si.", "Context.", "Parameters[\"assemblypath\"] = path;\n\n IDictionary stateSaver = new Hashtable();\n si.", "Install(stateSaver);\n}\n\nstatic void UninstallService(string name)\n{\n ServiceInstaller si = new ServiceInstaller();\n ServiceProcessInstaller spi = new ServiceProcessInstaller();\n si.", "Parent = spi;\n si.", "ServiceName = name;\n\n // update this if you want a different log\n si.", "Context = new InstallContext(\"uninstall.log\", null);\n si.", "Uninstall(null);\n}\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "Environmental factors modulating the extent of impact in coastal invasions: The case of a widespread invasive caprellid (Crustacea: Amphipoda) in the Iberian Peninsula.", "\nUnderstanding the respective roles of environment and interspecific interactions in shaping species' distributions is a critical aspect in determining the potential impacts of newcomer species on occupied habitats. ", "The invasive caprellid amphipod Caprella scaura has successfully spread along southern Europe in a short time period, coinciding with a decline in the population of an ecologically similar congener, Caprella equilibra. ", "To understand the mechanisms underlying the establishment success of this aggressive species and its potential role in shaping C. equilibra's distribution, we analyze the biotic and abiotic factors involved in the patterns of distribution and co-occurrence of both species along the Iberian Peninsula and northern Africa. ", "Our analyses support that C. scaura is competitively displacing C. equilibra from the study area, but also point out the critical role of salinity and temperature in modulating this interaction and limiting the invasive success of C. scaura on the Iberian North-Atlantic coast." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "[Polymorphism of glutathion-S-transferase genes in patients with prostatic cancer].", "\nThe study of polymorphic variants of GSTT1, GSTM1 and GSTP1 genes from 61 patients with prostatic cancer (PC) has shown that incidence of 0/0 genotype GSTT1 and GSTM1 in PC patients was significantly higher of that in healthy men (n = 100) (34.4 and 15% in p = 0.007 and 60.7 and 43% in p = 0.04, respectively). ", "PC risk in carriers of a GSTT1 deletion form was 2.97, CI95%--1.3-6.84, GSTM1--2.04 in CI95% 1.02-4.1. ", "The analysis of combinations of pathological genotypes of xenobiotic biotransformation enzymes has demonstrated that 89.8% PC patients have a mutation in one of the genes GSTT1, GSTM1 or GSTP1." ]

{ "pile_set_name": "PubMed Abstracts" }

[ 0.0007460520719178021, 0.0006432603113353252, 0.0009292119066230953, 0.0006535093416459858 ]

[ "Comparison of saline incubation and pepsin digestion as methods for recovering Ostertagia circumcincta larvae from the abomasum of sheep.", "\nIncubation in saline at 37 degrees C was compared with pepsin digestion as a quantitative method for recovering larval stages of Ostertagia circumcincta from the abomasal mucosa of sheep. ", "Experiments showed that the techniques were of roughly similar efficiency even when the larvae were arrested but that the saline method was preferable because worm searching was easier and quicker." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ ";\n;\tSharp OZ family functions\n;\n;\tported from the OZ-7xx SDK by by Alexander R. Pruss\n;\tby Stefano Bodrato - Oct. 2003\n;\n;\n;\tvoid ozscroll(unsigned numbytes);\n;\n;\n; ------\n; $Id: ozscroll.asm,v 1.3 2016-06-28 14:48:17 dom Exp $\n;\n\n SECTION code_clib\n\tPUBLIC\tozscroll\n\tPUBLIC\t_ozscroll\n\t\n\tEXTERN\tozactivepage\n\tEXTERN\trestore_a000\n\n\nozscroll:\n_ozscroll:\n pop hl\n exx\n\n ld de,(ozactivepage)\n\n ld a,e\n\tout\t(3),a\n ld a,d\n\tout\t(4),a\n\n ld hl,0a000h\n\n pop bc\n push bc\n\n add hl,bc\n\n ex de,hl\n\n ld hl,2400\n sbc hl,bc\n ld c,l\n ld b,h ;; length of move\n\n ex de,hl\n\n ld de,0a000h\n\n ldir\n\n call restore_a000\n\n exx\n jp (hl)\n\n" ]

{ "pile_set_name": "Github" }

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[ "Artie Lurie\n\nLibertarian for Florida House, District 90\n\nCandidate questionnaire from the Sun Sentinel\n\nSeptember 20, 2016\n\nI received a questionnaire from the Editorial Board of the Sun Sentinel newspaper ahead of my interview so they can determine who they will endorse. ", "Here is a copy of the questionnaire with my answers.", "\n\nIssues:\nWhy are you running for this office?", "\nI’m running because I want people to know that there is an alternative to what we’ve been getting forced upon us by our state government. ", "That our government is way too big, and through the benefits of liberty, we can have much smaller, and less intrusive government than what we have today.", "\nWhat is your vision?", "\nI believe that it is possible to have a state government that isn't involved in every aspect of our lives. ", "We need to focus on getting government out of all the areas where it has made such a complete mess and wasted our tax dollars. ", "I will support any legislation that will achieve this goal. ", "I believe we can have a state government that doesn't collect one dime in sales tax or corporate tax, and would still be able to perform the functions outlined in the constitution, so we can keep the money we earn and be able to live our lives according to one’s own values.", "\nBriefly, share your views on these issues:\n● The performance of Gov. Rick Scott -\nHe promises cutting taxes while endorsing bigger government every year since he’s been in office.", "\n● The challenges facing Florida’s economy -\nThe threat of increasing regulations and higher taxes cripple the growth of businesses and the pocketbooks of Floridians.", "\n● The top issue in public (government controlled) education -\nThe only way to make it possible for every child to get a good education is to get the politicians out of the schooling business and put it back in the hands of parents and teachers. ", "Let's repeal the sales tax and cut property taxes in half so parents can use the money for the school they choose.", "\n● Do you believe government closest to the people works best, or are contentious statewide issues best handled via state preemption bills?", "\nI believe in the motto “That government is best which governs least.”", "\n● What can be done to curb the rising cost of property insurance? ", "Government intrusion in the insurance business has caused the cost of insurance to skyrocket to a point where some people can’t even afford to get it. ", "Repealing state insurance laws will make it much more affordable and a lot more accessible to Floridians.", "\n\n● What do you see as the next step in health care reform?", "\n\nThe only way we can get affordable and accessible healthcare is by getting the government completely out of healthcare.", "\n\n● What should be done about the problems in our prisons?", "\n\nOur prisons are overflowing with people who committed non-violent drug offenses and other victimless crimes, while the violent criminals get off with plea bargains and early releases. ", "Let’s put an end to the drug war and other victimless crimes so we can keep the real criminals behind bars.", "\n\n● Do you support full funding for Enterprise Florida?", "\n\nNo. ", "Enterprise Florida is just another big government boondoggle forcing taxpayers to pay for a program where the politicians can dole out money to the people with the most political influence. ", "Where the only benefits are to the politicians and not the taxpayers.", "\n\n● What role should state government play in addressing the violence in our cities, with the shootings of police officers and police shootings of black men?", "\n\nThe government should start by upholding the constitutional rights of all Floridians and repealing all of the unnecessary victimless crimes which are the pretext for all encounters which ultimately leads to unnecessary violence.", "\n\n● What should be done about the damage caused by releases from Lake Okeechobee and who should take the lead?", "\n\nThe Government has proven time and again that it cannot keep the environment clean. ", "The government is the biggest polluter of our rivers, lakes and streams. ", "The lake should be sold or placed into private interests who would be much more accountable to consumers and ensure that it stays clean without using taxpayers money.", "\n\n● In Florida, there is a presumption of government openness, meaning that all records and meetings are presumed open to the public absent a specific statutory exemption. ", "During the 2016 legislative session, an exemption for trade secrets was expanded to include financial information, including the amount government pays its contractors. ", "Do you think the amount of money government pays its contractors should be secret? ", "Are there government records currently subject to disclosure that you believe should be exempt?", "\n\nTransparency should be paramount in our government. ", "The big government politicians forget that they are taking money from the taxpayers, and the taxpayers have a right to know where the money is being spent.", "\n\n● Do you believe the climate is changing? ", "If so, what should the state do now?", "\n\nI do not believe that the climate is changing in a significant way where the politicians can come up with any program that will work to solve it.", "\n\n● Do you support a controlled expansion of casino gambling? ", "I do not believe that a company should need to get a permission slip from the government in order to open a casino. ", "Free people should be allowed to conduct business in any way they choose as long as they don’t cause harm to someone else.", "\n\n● Do you support the medical marijuana amendment? ", "Yes. ", "Medical marijuana is a step in the right direction. ", "But it doesn’t change the fact that the drug war is a complete failure and we should take steps to end drug prohibition in Florida.", "\n\n● Do you believe reforms are needed in the area of gun control? ", "Absolutely not. ", "I believe that Gun control laws do the exact opposite of the very thing it is trying to accomplish, to keep people safe! ", "Gun regulations disarm innocent civilians while doing nothing to prevent the violent criminals from committing atrocities and mass shootings.", "\n\n● Do you believe reforms are needed around the issue of abortion? ", "I do not believe that the government should have the authority to decide who should live or die, and that goes for the unborn as well.", "\n\n● Do you support red-light cameras? ", "No. ", "Red light cameras are a scam to make profit for the government under the guise of safety. ", "Red light cameras are making intersections less safe, and the state laws governing their use, in my opinion, are unconstitutional.", "\n\n● Do you support fracking? ", "Any type of method to extract natural resources on private property should not be the concern of politicians. ", "Most of the problems with the issue of fracking occur on government controlled land. ", "Let’s get the government out the land owning business.", "\n\nThe campaign:\n\nIs there anything you want us to know about your opponent?", "\n\nMy opponent does not represent the people that live in our district. ", "She says she claims to represent our values, but in reality represents only her own agenda and of the special interests who support making government bigger and more intrusive into our lives. ", "I resent the fact she is skirting election laws by renting an apartment in the district to get on the ballot.", "\n\nWhy should voters vote for you?", "\n\nAs a Libertarian, I am committed to the principles of freedom, personal responsibility, and limited government. ", "Floridians are sick to death of supporting the waste and corruption going on in Tallahassee at the expense of their own families. ", "It's time to elect a candidate who believes in your freedom and is committed to drastically reduce the size of our state government." ]

{ "pile_set_name": "Pile-CC" }

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[ "Insights from latent partition analysis into categories inherent in wellness-illness.", "\nHow health-disease is perceived or conceptualized is important for nursing research. ", "There is increasing evidence that individual representations are important in constructing the experience of health-disease. ", "What is the personal saliency of health-disease for the individual? ", "To explore the patterns of meaning inherent in health-disease, a card sort was undertaken among 15 healthy individuals and 15 individuals with chronic renal disease. ", "Both groups were given 28 cards to sort twice: once for when they felt 'well' and again for when they felt 'ill'. ", "The theoretical basis underlying the items of the card sort was a model of wellness-illness being developed. ", "Latent partition analysis was used to cluster the concepts from each data set followed by multi-dimensional scaling to analyse the structure of the intercategory probability estimates. ", "A possible unidimensional pattern of meaning (harmony) emerged for the 'well' data and a two-dimensional pattern (disharmony and optimism) for the 'ill' data. ", "This represents a preliminary step in the development of a theoretical model that would permit assessment of the meaning of health-disease for the individual." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Q:\n\nHow does HTTP specify the end of a response for Transfer-encoding: chunked?", "\n\nFrom this wikipedia example\n a typical HTTP response may look like this \nHTTP/1.1 200 OK\nDate: Mon, 23 May 2005 22:38:34 GMT\nContent-Type: text/html; charset=UTF-8\nContent-Encoding: UTF-8\nContent-Length: 138\nLast-Modified: Wed, 08 Jan 2003 23:11:55 GMT\nServer: Apache/1.3.3.7 (Unix) (Red-Hat/Linux)\nETag: \"3f80f-1b6-3e1cb03b\"\nAccept-Ranges: bytes\nConnection: close\n\n<html>\n<head>\n <title>An Example Page</title>\n</head>\n<body>\n Hello World, this is a very simple HTML document.", "\n</body>\n</html>\n\nIn this case because of this header:\nConnection: close\n\nthe client will know that the HTTP body has finished, because the TCP connection will be closed, Say for example Connection: close wasn't present, the client would still know the HTTP body was finished because this header:\nContent-Length: 138\n\nIs saying, once you get 138 bytes, we're done here. ", "But in the case where neither of these headers are used and instead the server sends Transfer-encoding: chunked how does a browser know when a response has completed, so that it can move onto other things?", "\n\nA:\n\nFrom Wikipedia:\n\nThe chunked keyword in the Transfer-Encoding header is used to\n indicate chunked transfer. ", "Each chunk is preceded by its size. ", "The\n transmission ends when a zero-length chunk is encountered.", "\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "Direct ELISA.", "\nFirst described by Engvall and Perlmann, the enzyme-linked immunosorbent assay (ELISA) is a rapid and sensitive method for detection and quantitation of an antigen using an enzyme-labeled antibody. ", "Besides routine laboratory usage, ELISA has been utilized in medical field and food industry as diagnostic and quality control tools. ", "Traditionally performed in 96-well or 384-well polystyrene plates, the technology has expanded to other platforms with increase in automation. ", "Depending on the antigen epitope and availability of specific antibody, there are variations in ELISA setup. ", "The four basic formats are direct, indirect, sandwich, and competitive ELISAs. ", "Direct ELISA is the simplest format requiring an antigen and an enzyme-conjugated antibody specific to the antigen. ", "This chapter describes the individual steps for detection of a plate-bound antigen using a horseradish peroxidase (HRP)-conjugated antibody and luminol-based enhanced chemiluminescence (ECL) substrate. ", "The methodological approach to optimize the assay by chessboard titration is also provided." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "package one\n" ]

{ "pile_set_name": "Github" }

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[ "[Changes in the reactive properties of the body under anesthesia against a background of aspartic acid premedication].", "\nIt was shown in acute experiments on mice and dogs that aspartic acid and its sodium salt administered 15-30 minutes prior to ether and hexenal anesthesia prolong the animals' life span by 42.2-99.9% compared to control given an isotonic solution of sodium chloride. ", "The stabilizing action of the aspartate is accounted for by its effect on the reactive properties of the central nervous system, particularly on the new cortex, old cortex, archicortex and circulation. ", "This effect is also associated with the increased oxygen tension in brain tissues, with increased heart contractions and slowing down of their rhythm." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "1. ", "Field of the Invention\nThe present invention relates to technology for embedding/reading additional information such as copyright information and editing information into/from various data such as image data including moving images and still images, music data, and audio data. ", "More particularly, the present invention relates to a digital watermark embedding (may also be referred to as “data hiding”) apparatus and method and a computer program for embedding a digital watermark as additional information imperceptible to normal visual or auditory observation into image data or audio data.", "\n2. ", "Description of the Related Art\nAs digital technology progresses, digital recorder/players that do not cause deterioration in image quality and sound quality due to repetitive recording and playing have become widely used. ", "At the same time, various types of digital content such as images and music content have become distributable through media such as digital VCRs (videocassette recorders), DVDs (digital versatile disks), and CD (compact discs) and networks.", "\nUnlike analog recording and playing, digital recording and playing do not cause deterioration of data even when recording and playing are repeatedly performed. ", "As a result, the same quality as that of the original data can be maintained. ", "The widely spread use of digital recording and playing technology allows for an excess of illegal copying. ", "This is a serious problem in view of copyright protection.", "\nIn order to protect digital content against copyright infringement by illegal copying, the following scheme is devised. ", "Specifically, the scheme involves adding copy control information to digital content, reading the copy control information when the content is recorded or played, and performing processing in accordance with the read copy control information, thus preventing illegal copying.", "\nThere are various content control schemes. ", "A typical scheme employs, for example, a copy generation management system (CGMS). ", "For an analog video signal (referred to as CGMS-A), the CGMS scheme superimposes copy control information, that is, two bits of 20-bit additional information to be superimposed on an effective video portion of a specific horizontal period, e.g., the 20th horizontal period in case of an NTSC signal, in a vertical blanking interval of a luminance signal. ", "For a digital video signal (referred to as CGMS-D), the CGMS scheme transmits the digital video signal (digital video data) and two bits of copy control information serving as additional information to be inserted into the digital video data.", "\nThe two-bit information used in the CGMS scheme (referred to as CGMS information) has the following meanings: “00” means that copying is permitted, “10” means that one copying is permitted (one generation is permitted), and “11” means that copying is prohibited (copying is absolutely prohibited).", "\nThe CGMS scheme described above is an example of a typical copy control scheme. ", "There are other schemes for protecting content copyright. ", "For example, digital broadcasting performed by a broadcast station stores a digital copy control descriptor in program schedule information, that is, service information (SI), included in a transport stream (TS) packet forming digital data. ", "When recording data received by a receiver in a recorder, copy generation control in accordance with the descriptor is performed.", "\nThe foregoing control information is added as bit data to the header of content and cannot completely eliminate possible illegal alteration of the added data. ", "Digital watermarking is advantageous in eliminating the possibility of data alteration. ", "When content (image data or audio data) is normally played, a digital watermark is imperceptible to vision and auditory senses. ", "Digital watermarks can only be detected/embedded by execution of a specific algorithm or processing by a specific device. ", "A digital watermark is detected when content is processed by a receiver or recorder/player, and processing in accordance with the digital watermark is performed, thus making more reliable control possible.", "\nInformation that can be embedded in content by digital watermarking includes not only the above-described copy control information, but also various information such as content copyright information, content processing information, content format information, content editing information, and content playing information.", "\nInformation such as a digital watermark that is directly superimposed on an information signal has a strong alteration resistance and is expected to serve as secure additional information. ", "In digital watermarking technology, the quality of a watermarked image may become a problem. ", "Since digital watermarking performs direct signal processing of data to be watermarked (e.g., an image), the quality of the image may deteriorate or the statistic characteristics of the image may be biased, leading to deterioration of the original image. ", "Digital watermarking that does not cause adverse effects on content is highly demanded.", "\nAn example of a known method for preventing deterioration of original data involves setting digital watermark embedding parameters in individual data areas in the original data such as an image to be watermarked, and subsequently, again setting a digital watermark superimposing level adjusting parameter (global parameter) in the entire data area taking into consideration the deterioration in image quality.", "\nThere are various methods for controlling embedding of digital watermarks. ", "In one method, an amount embedded is increased by utilizing the characteristics of an image. ", "In another method, a digital watermark is embedded at the minimum embedding level so that priority may be given to image quality. ", "There is not yet a digital watermark embedding algorithm, commonly applicable to all information signals such as image or audio data to be watermarked, for both achieving a sufficient digital watermark detecting accuracy and preventing deterioration in image quality. ", "By embedding a digital watermark, the quality of a data signal such as image or audio data may deteriorate, or the reliability of detecting an embedded digital watermark may be reduced.", "\nIn the foregoing digital watermark superimposing level controlling method using the global parameter, a first algorithm for determining initial embedding parameters determines substantially an optimal level of superimposing a digital watermark in each data area. ", "Subsequently, the final adjustment involving the global parameter is performed. ", "The optimal parameters initialized by the first algorithm in individual data areas are forcedly changed by setting the global parameter that takes into consideration deterioration in image quality.", "\nSetting the global parameter may not be favorable in view of digital watermark detection. ", "In other words, the initial optimal parameters are set so as to maintain the superimposing level that ensures digital watermark detection. ", "When the initial optimal parameters are changed by setting the global parameter, the sufficient detection level may not be achieved." ]

{ "pile_set_name": "USPTO Backgrounds" }

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[ "St. Bonaventure University senior Christopher Radey says he had to buy a ticket for a bus trip to watch the men’s basketball team play in the Atlantic 10 championship game in Atlantic City, N.J.\n\n“It was too monumental of a moment for our school to not be there cheering on with all my peers,” explains Radey.", "\n\nSt. Bonaventure earned its first conference title and first NCAA tournament appearance in nine years after defeating Xavier 67-56 on Sunday. ", "While one bus of students saw the historic victory live, Radey unexpectedly watched from an Applebee’s with other passionate fans.", "\n\nTwo Covered Wagon Tours buses left campus at 4 a.m. The second bus’ driver noticed something leaking from the first bus as it approached a Great Bend, Pa., McDonald’s three hours later.", "\n\nGraduate student Kathryn Klimek, the first bus’ chaperone, says after both buses stopped, mechanics worked to repair an oil leak on the first bus. ", "Klimek says the woman who helped reserve the buses made phone calls to arrange another bus.", "\n\nRadey says he and some passengers wondered if the second bus would leave, comparing the situation to a movie.", "\n\n“We decided that the two buses were exactly like those two boats full of people in The Dark Knight,” Radey remembers. “", "People were passing judgment on each other, deciding who should stay, who should go.”", "\n\nAs mechanics worked, the second bus left, abandoning 41 disappointed students.", "\n\nSenior Lisa Malmgren says passengers desperately tried calling bus companies.", "\n\n“It’s hard to get ahold of any business at 8 a.m. on a Sunday,” says Malmgren. “", "We just had to accept our fate.”", "\n\nKlimek says she held out hope until 10 a.m.\n\n“At that point we would not have made it there for any of the first half and we would be cutting it close to even make for the second half if we didn’t hit one minute of traffic,” she explains. “", "I was very sad for the group of students.”", "\n\nAt 10:30 a.m., another bus arrived and drove the group back towards campus. ", "It stopped at the Hornell, N.Y. Applebee’s parking lot about 20 minutes before the 1 p.m. tipoff. ", "The passengers entered the restaurant, creating a smaller version of St. Bonaventure’s home court, the Reilly Center.", "\n\n“We took over Applebee’s and cheered like crazy,” says senior Kate Burt. “", "In true Bona fashion, we made the best out of a crappy situation.”", "\n\n“The staff was very accommodating,” says senior Sam Wilson. “", "It was a bonus to watch the game in a place with beer on tap and pretty good food.”", "\n\nKlimek says she agrees the students handled the situation extremely well.", "\n\n“The bus was full of the best sports I could have asked for,” states Klimek.", "\n\n“I loved the experience we had and it will forever make a wonderful story that I can tell people,” declares Radey.", "\n\nAs one of 25 schools to have both its men’s and women’s basketball teams in the NCAA tournament this year, the students say they see the teams’ successes as a significant achievement.", "\n\n“Students from bigger schools can’t understand what it’s like,” says Malmgren. “", "We’re such a small school, but so passionate. ", "I think it’s about time to show people what it means to be a Bonnie.”", "\n\n“I couldn’t imagine being anywhere but Bonaventure for this experience,” says Klimek.", "\n\n“I’ve never felt more proud of this school.” ", "marvels Wilson.", "\n\nThe men’s team takes on Florida State Friday at 2:45 p.m. The women’s team plays Florida Gulf Coast Sunday at 2:30 p.m.\n\nPhoto credit:\n\nBryan Weber" ]

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[ "package winrm\n\nimport (\n\t\"bytes\"\n\t\"crypto/tls\"\n\t\"crypto/x509\"\n\t\"fmt\"\n\t\"io\"\n\t\"net/http\"\n\n\t\"github.com/masterzen/winrm/soap\"\n)\n\ntype Client struct {\n\tParameters\n\tusername string\n\tpassword string\n\tuseHTTPS bool\n\turl string\n\thttp HttpPost\n\ttransport http.", "RoundTripper\n}\n\n// NewClient will create a new remote client on url, connecting with user and password\n// This function doesn't connect (connection happens only when CreateShell is called)\nfunc NewClient(endpoint *Endpoint, user, password string) (client *Client, err error) {\n\tparams := DefaultParameters()\n\tclient, err = NewClientWithParameters(endpoint, user, password, params)\n\treturn\n}\n\n// NewClient will create a new remote client on url, connecting with user and password\n// This function doesn't connect (connection happens only when CreateShell is called)\nfunc NewClientWithParameters(endpoint *Endpoint, user, password string, params *Parameters) (client *Client, err error) {\n\ttransport, err := newTransport(endpoint)\n\n\tclient = &Client{\n\t\tParameters: *params,\n\t\tusername: user,\n\t\tpassword: password,\n\t\turl: endpoint.url(),\n\t\thttp: Http_post,\n\t\tuseHTTPS: endpoint.", "HTTPS,\n\t\ttransport: transport,\n\t}\n\n\tif params.", "TransportDecorator !", "= nil {\n\t\tclient.transport = params.", "TransportDecorator(transport)\n\t}\n\treturn\n}\n\n// newTransport will create a new HTTP Transport, with options specified within the endpoint configuration\nfunc newTransport(endpoint *Endpoint) (*http.", "Transport, error) {\n\ttransport := &http.", "Transport{\n\t\tTLSClientConfig: &tls.", "Config{\n\t\t\tInsecureSkipVerify: endpoint.", "Insecure,\n\t\t},\n\t}\n\n\tif endpoint.", "CACert !", "= nil && len(*endpoint.", "CACert) > 0 {\n\t\tcertPool, err := readCACerts(endpoint.", "CACert)\n\t\tif err !", "= nil {\n\t\t\treturn nil, err\n\t\t}\n\n\t\ttransport.", "TLSClientConfig.", "RootCAs = certPool\n\t}\n\n\treturn transport, nil\n}\n\nfunc readCACerts(certs *[]byte) (*x509.CertPool, error) {\n\tcertPool := x509.NewCertPool()\n\n\tif !", "certPool.", "AppendCertsFromPEM(*certs) {\n\t\treturn nil, fmt.", "Errorf(\"Unable to read certificates\")\n\t}\n\n\treturn certPool, nil\n}\n\n// CreateShell will create a WinRM Shell, which is the prealable for running\n// commands.", "\nfunc (client *Client) CreateShell() (shell *Shell, err error) {\n\trequest := NewOpenShellRequest(client.url, &client.", "Parameters)\n\tdefer request.", "Free()\n\n\tresponse, err := client.sendRequest(request)\n\tif err == nil {\n\t\tvar shellId string\n\t\tif shellId, err = ParseOpenShellResponse(response); err == nil {\n\t\t\tshell = &Shell{client: client, ShellId: shellId}\n\t\t}\n\t}\n\treturn\n}\n\n// NewShell will create a new WinRM Shell for the given shellID\nfunc (client *Client) NewShell(shellID string) *Shell {\n\treturn &Shell{client: client, ShellId: shellID}\n}\n\nfunc (client *Client) sendRequest(request *soap.", "SoapMessage) (response string, err error) {\n\treturn client.http(client, request)\n}\n\n// Run will run command on the the remote host, writing the process stdout and stderr to\n// the given writers. ", "Note with this method it isn't possible to inject stdin.", "\nfunc (client *Client) Run(command string, stdout io.", "Writer, stderr io.", "Writer) (exitCode int, err error) {\n\tshell, err := client.", "CreateShell()\n\tif err !", "= nil {\n\t\treturn 0, err\n\t}\n\tvar cmd *Command\n\tcmd, err = shell.", "Execute(command)\n\tif err !", "= nil {\n\t\treturn 0, err\n\t}\n\tgo io.", "Copy(stdout, cmd.", "Stdout)\n\tgo io.", "Copy(stderr, cmd.", "Stderr)\n\tcmd.", "Wait()\n\tshell.", "Close()\n\treturn cmd.", "ExitCode(), cmd.err\n}\n\n// Run will run command on the the remote host, returning the process stdout and stderr\n// as strings, and using the input stdin string as the process input\nfunc (client *Client) RunWithString(command string, stdin string) (stdout string, stderr string, exitCode int, err error) {\n\tshell, err := client.", "CreateShell()\n\tif err !", "= nil {\n\t\treturn \"\", \"\", 0, err\n\t}\n\tdefer shell.", "Close()\n\tvar cmd *Command\n\tcmd, err = shell.", "Execute(command)\n\tif err !", "= nil {\n\t\treturn \"\", \"\", 0, err\n\t}\n\tif len(stdin) > 0 {\n\t\tcmd.", "Stdin.", "Write([]byte(stdin))\n\t}\n\tvar outWriter, errWriter bytes.", "Buffer\n\tgo io.", "Copy(&outWriter, cmd.", "Stdout)\n\tgo io.", "Copy(&errWriter, cmd.", "Stderr)\n\tcmd.", "Wait()\n\treturn outWriter.", "String(), errWriter.", "String(), cmd.", "ExitCode(), cmd.err\n}\n\n// Run will run command on the the remote host, writing the process stdout and stderr to\n// the given writers, and injecting the process stdin with the stdin reader.", "\n// Warning stdin (not stdout/stderr) are bufferized, which means reading only one byte in stdin will\n// send a winrm http packet to the remote host. ", "If stdin is a pipe, it might be better for\n// performance reasons to buffer it.", "\nfunc (client *Client) RunWithInput(command string, stdout io.", "Writer, stderr io.", "Writer, stdin io.", "Reader) (exitCode int, err error) {\n\tshell, err := client.", "CreateShell()\n\tif err !", "= nil {\n\t\treturn 0, err\n\t}\n\tdefer shell.", "Close()\n\tvar cmd *Command\n\tcmd, err = shell.", "Execute(command)\n\tif err !", "= nil {\n\t\treturn 0, err\n\t}\n\tgo io.", "Copy(cmd.", "Stdin, stdin)\n\tgo io.", "Copy(stdout, cmd.", "Stdout)\n\tgo io.", "Copy(stderr, cmd.", "Stderr)\n\tcmd.", "Wait()\n\treturn cmd.", "ExitCode(), cmd.err\n}\n" ]

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[ "Her asshole stretches open as she shoots cum with every prolapse! ", "And then see her insert a fat dildo deep into her asshole. ", "Then watch her take a long beige brutal dildo! ", "Watch her gape her holes in this hot brutal anal and vaginal destruction! ", "Brithney Brithney is dripping wet before she even starts fucking these two massive brutal dildos." ]

{ "pile_set_name": "Pile-CC" }

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[ "These are the writings of a middle-aged guy trying to be as moderate as I possibly can. ", "Here is where I put down in words the stuff that comes pourin' outta my head, and do my bitchin' and complainin' about the government or whatever else pisses me off. ", "Everything and everyone are fair game. ", "These are my ramblings, rantings and ravings with the occasional twist of humor and/or sarcasm.", "\nAll this while doing my part to fight the Fascists.", "\n\nMy Liberal Identity:\n\nYou are a Working Class Warrior, also known as a blue-collar Democrat. ", "You believe that the little guy is getting screwed by conservative greed-mongers and corporate criminals, and you’re not going to take it anymore.", "\n\nLocal Yocal News\n\nQUOTES TO CONSIDER\n\nWhen the people fear their government, there is tyranny; when the government fears the people, there is liberty. -- ", "Thomas Jefferson\n\nA nation that is afraid to let its people judge the truth and falsehood in an open market is a nation that is afraid of its people. -- ", "John F. Kennedy\n\nLoyalty to the country always. ", "Loyalty to the government when it deserves it. -- ", "Mark Twain\n\nWhen fascism comes to America, it will be wrapped in the flag, carrying a cross. -- ", "Sinclair Lewis\n\nThere is danger from all men. ", "The only maxim of a free government ought to be to trust no man living with power to endanger the public liberty. -- ", "John Adams\n\nMen never do evil so completely and cheerfully as when they do it from a religious conviction. -- ", "Blaise Pascal\n\nThe spirit of resistance to government is so valuable on certain occasions, that I wish it to be always kept alive. ", "It will often be exercised when wrong, but better so than not to be exercised at all. ", "I like a little rebellion now and then. ", "It is like a storm in the atmosphere. -- ", "Thomas Jefferson to Abigail Adams, 1787.", "\n\nComments from all readers are welcome and appreciated. ", "Open discussion is allowed, trolls are not. ", "As I am the editor of this blog, I am the one who makes the decision on what may be posted in the comments section. ", "Free speech is a wonderful thing, but comments from trolls will be removed...and I decide who a troll is here." ]

{ "pile_set_name": "Pile-CC" }

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[ "Timeline of the 1996 Atlantic hurricane season\n\nThe 1996 Atlantic hurricane season had 13 named storms, of which 9 became hurricanes and 6 became major hurricanes (hurricanes that are classified as Category 3 or higher). ", "These major hurricanes were Bertha, Edouard, Fran, Hortense, Isidore, and Lili. ", "This timeline documents all the storm formations, strengthening, weakening, landfalls, extratropical transitions, as well as dissipation. ", "The timeline also includes information that was not operationally released, meaning that information from post-storm reviews by the National Hurricane Center, such as information on a storm that was not operationally warned on, has been included. ", "The season officially began on June 1, 1996, and ended on November 30 that same year.", "\n\nThe season's most destructive storms were Hurricane Cesar, Hurricane Fran, and Hurricane Hortense. ", "Hurricane Cesar (later known as Hurricane Douglas in the Eastern Pacific basin) was the deadliest storm of the season; it killed at least 51 people and caused severe damage in northern Colombia and southern Central America. ", "Hurricane Fran caused $3.2 billion (1996 USD; $4 billion 2009 USD) worth of damage in the United States, mostly in North Carolina, and killed 26 people. ", "Hurricane Hortense dropped torrential rainfall on southwestern Puerto Rico and the eastern Dominican Republic, killing 21 people and leaving behind $127 million (1996 USD; $174 million 2009 USD) in damage. ", "All three storms had their names retired by the World Meteorological Organization in the spring of 1997, and were replaced with Cristobal, Fay, and Hanna for the 2002 season, respectively.", "\n\nTimeline of storms\n\nJune\nJune 1\n12:00 am EDT (0400 UTC) – The 1996 Atlantic hurricane season officially begins.", "\nJune 17\n\n2:00 pm EDT (1600 UTC) – Tropical depression One forms near eastern Grand Bahama.", "\nJune 18\n8:00 pm EDT (0000 UTC June 19) – Tropical depression One strengthens into Tropical Storm Arthur.", "\nJune 19\n8:00 pm EDT (0000 UTC June 20) – Tropical Storm Arthur makes landfall near Cape Lookout on the North Carolina Outer Banks with 40 mph (65 km/h) winds.", "\nJune 20\n8:00 am EDT (1200 UTC) – Tropical Storm Arthur weakens into a tropical depression.", "\nJune 21\n8:00 am EDT (1200 UTC) – Tropical Depression Arthur becomes extratropical about 350 nautical miles (400 miles, 645 km) north-northeast of Bermuda.", "\n\nJuly\nJuly 4\n8:00 pm AST (0000 UTC July 5) – Tropical depression Two forms in the south-central Atlantic basin.", "\nJuly 5\n8:00 am AST (1200 UTC) – Tropical depression Two strengthens into Tropical Storm Bertha.", "\n\nJuly 7\n2:00 pm AST (1800 UTC) – Tropical Storm Bertha strengthens into Hurricane Bertha as it moves across the Leeward and Virgin Islands.", "\nJuly 9\n2:00 am EDT (0600 UTC) – Hurricane Bertha reaches Category 3 intensity.", "\n2:00 pm EDT (1800 UTC) – Hurricane Bertha weakens into a Category 2 hurricane.", "\nJuly 10\n2:00 am EDT (0600 UTC) – Hurricane Bertha weakens into a Category 1 hurricane.", "\nJuly 12\n8:00 am EDT (1200 UTC) – Hurricane Bertha regains Category 2 intensity.", "\n4:00 pm EDT (2000 UTC) – Hurricane Bertha makes landfall between Wrightsville and Topsail Beaches in North Carolina with 105 mph (170 km/h) winds.", "\n8:00 pm EDT (0000 UTC July 13) – Hurricane Bertha weakens into a Category 1 hurricane.", "\nJuly 13\n2:00 am EDT (0600 UTC) – Hurricane Bertha weakens into a tropical storm.", "\nJuly 14\n8:00 am EDT (1200 UTC) – Tropical Storm Bertha becomes extratropical near New Brunswick.", "\nJuly 24\n\n2:00 pm AST (1800 UTC) – Tropical depression Three forms just north of Isla Margarita.", "\nJuly 25\n8:00 am EDT (1200 UTC) – Tropical depression Three strengthens into Tropical Storm Cesar.", "\nJuly 27\n8:00 am EDT (1200 UTC) – Tropical Storm Cesar strengthens into Hurricane Cesar.", "\nJuly 28\n12:00 am EDT (0400 UTC) – Hurricane Cesar makes landfall just north of Bluefields, Nicaragua, with 85 mph (135 km/h) winds.", "\n8:00 am EDT (1200 UTC) – Hurricane Cesar weakens into a tropical storm.", "\n7:00 pm CDT (0000 UTC July 29) – Tropical Storm Cesar completes its trek across Nicaragua, and becomes Tropical Storm Douglas in the Eastern Pacific.", "\n\nAugust\nAugust 19\n\n2:00 am EDT (0600 UTC) – Tropical depression Four forms in the northwestern Caribbean Sea.", "\n2:00 pm EDT (1800 UTC) – Tropical depression Four strengthens into Tropical Storm Dolly.", "\n2:00 pm AST (1800 UTC) – Tropical depression Five forms about 100 miles (160 km) west of the Senegal/Guinea-Bissau border.", "\nAugust 20\n12:30 pm CDT (1730 UTC) – Tropical Storm Dolly strengthens into Hurricane Dolly as it makes its first landfall near Punta Herrero, Mexico with 75 mph (120 km/h) winds.", "\n7:00 pm CDT (0000 UTC August 21) – Hurricane Dolly weakens into a tropical storm.", "\nAugust 21\n7:00 am CDT (1200 UTC) – Tropical Storm Dolly weakens into a tropical depression.", "\n7:00 pm CDT (0000 UTC August 22) – Tropical Depression Dolly regains tropical storm status.", "\nAugust 22\n2:00 am AST (0600 UTC) – Tropical depression Five, which formed about 100 miles (160 km) west of the Senegal/Guinea-Bissau border on August 19, strengthens into Tropical Storm Edouard.", "\nAugust 23\n7:00 am CDT (1200 UTC) – Tropical Storm Dolly regains hurricane status as it makes its second landfall midway between Tampico, Tamaulipas, and Tuxpan, Veracruz, with 80 mph (130 km/h) winds.", "\n8:00 am AST (1200 UTC) – Tropical Storm Edouard strengthens into Hurricane Edouard.", "\n8:00 am AST (1200 UTC) – Tropical depression Six forms just southeast of the Cape Verde Islands.", "\n1:00 pm CDT (1800 UTC) – Hurricane Dolly weakens into a tropical storm.", "\n7:00 pm CDT (0000 UTC August 24) – Tropical Storm Dolly weakens into a tropical depression.", "\nAugust 24\n\n8:00 am AST (1200 UTC) – Hurricane Edouard reaches Category 2 intensity.", "\n2:00 pm AST (1800 UTC) – Hurricane Edouard reaches Category 3 intensity.", "\n8:00 pm AST (0000 UTC August 25) – Hurricane Edouard reaches Category 4 intensity.", "\nAugust 25\n12:00 am MDT (0600 UTC) – Tropical Depression Dolly dissipates over the eastern North Pacific Ocean.", "\n2:00 am AST (0600 UTC) – Hurricane Edouard reaches its peak intensity of 145 mph (230 km/h) winds and a pressure of 933 mbar, making it the strongest storm of the season.", "\n8:00 pm AST (0000 UTC August 26) – Tropical depression Seven forms south of the Cape Verde Islands.", "\nAugust 27\n8:00 am AST (1200 UTC) – Tropical depression Six strengthens into Tropical Storm Fran.", "\nAugust 28\n2:00 am AST (0600 UTC) – Tropical depression Seven strengthens into Tropical Storm Gustav.", "\n8:00 am AST (1200 UTC) – Hurricane Edouard weakens into a Category 3 hurricane.", "\nAugust 29\n8:00 am EDT (1200 UTC) – Hurricane Edouard regains Category 4 intensity.", "\n8:00 pm AST (0000 UTC August 30) – Tropical Storm Fran strengthens into Hurricane Fran.", "\nAugust 30\n2:00 pm AST (1800 UTC) – Hurricane Fran weakens into a tropical storm.", "\n8:00 pm EDT (0000 UTC August 31) – Hurricane Edouard weakens back into a Category 3 hurricane.", "\nAugust 31\n8:00 am AST (1200 UTC) – Tropical Storm Fran regains hurricane status.", "\n\nSeptember\n\nSeptember 1\n8:00 am EDT (1200 UTC) – Hurricane Edouard weakens into a Category 2 hurricane.", "\n2:00 pm AST (1800 UTC) – Tropical Storm Gustav weakens into a tropical depression.", "\n8:00 pm EDT (0000 UTC September 2) – Hurricane Edouard weakens into a Category 1 hurricane.", "\nSeptember 2\n2:00 am AST (0600 UTC) – Tropical Depression Gustav dissipates.", "\n8:00 pm EDT (0000 UTC September 3) – Hurricane Edouard weakens into a tropical storm.", "\nSeptember 3\n2:00 am EDT (0600 UTC) – Tropical Storm Edouard becomes extratropical about 100 miles (160 km) south-southeast of the Maine/Canada.", "\n8:00 am AST (1200 UTC) – Tropical depression Eight forms about midway between the northern Leeward Islands and Senegal.", "\n2:00 pm EDT (1800 UTC) – Hurricane Fran reaches Category 2 intensity.", "\nSeptember 4\n2:00 am EDT (0600 UTC) – Hurricane Fran reaches Category 3 intensity.", "\nSeptember 5\n8:30 pm EDT (0030 UTC September 6) – Hurricane Fran makes landfall near Cape Fear, North Carolina with 115 mph (185 km/h) winds.", "\nSeptember 6\n2:00 am EDT (0600 UTC) – Hurricane Fran rapidly weakens to a Category 1 hurricane.", "\n8:00 am EDT (1200 UTC) – Hurricane Fran weakens into a tropical storm.", "\n2:00 pm EDT (1800 UTC) – Tropical Storm Fran weakens into a tropical depression.", "\nSeptember 7\n2:00 am AST (0600 UTC) – Tropical depression Eight strengthens into Tropical Storm Hortense.", "\nSeptember 8\n8:00 pm EDT (0000 UTC September 9) – Tropical Depression Fran becomes extratropical in southern Ontario.", "\nSeptember 9\n\n2:00 am AST (0600 UTC) – Tropical Storm Hortense strengthens into Hurricane Hortense.", "\nSeptember 10\n2:00 am EDT (0600 UTC) – Hurricane Hortense makes its first landfall near Guánica, Puerto Rico with 80 mph (130 km/h) winds.", "\nSeptember 11\n8:00 am EDT (1200 UTC) – Hurricane Hortense reaches Category 2 intensity.", "\n8:00 pm EDT (0000 UTC September 12) – Hurricane Hortense reaches Category 3 intensity.", "\nSeptember 12\n8:00 am EDT (1200 UTC – Hurricane Hortense reaches Category 4 intensity.", "\nSeptember 13\n8:00 am EDT (1200 UTC) – Hurricane Hortense weakens into a Category 3 hurricane.", "\n8:00 pm EDT (0000 UTC September 14) – Hurricane Hortense weakens into a Category 2 hurricane.", "\nSeptember 14\n2:00 pm AST (1800 UTC) – Hurricane Hortense weakens into a Category 1 hurricane.", "\n11:00 pm AST (0300 UTC September 15) – Hurricane Hortense makes its second landfall in Nova Scotia with 80 mph (130 km/h) winds.", "\nSeptember 15\n8:00 am AST (1200 UTC) – Hurricane Hortense weakens into a tropical storm.", "\n2:00 pm AST (1800 UTC) – Tropical Storm Hortense becomes extratropical just south of Newfoundland.", "\nSeptember 24\n\n8:00 am AST (1200 UTC) – Tropical depression Nine forms about 150 miles (240 km) south of Santiago, Cape Verde.", "\nSeptember 25\n2:00 am AST (0600 UTC) – Tropical depression Nine strengthens into Tropical Storm Isidore.", "\nSeptember 26\n2:00 am AST (0600 UTC) – Tropical Storm Isidore strengthens into Hurricane Isidore.", "\nSeptember 27\n2:00 am AST (0600 UTC) – Hurricane Isidore reaches Category 2 intensity.", "\n8:00 pm AST (0000 UTC September 28) – Hurricane Isidore reaches Category 3 intensity.", "\nSeptember 28\n8:00 am AST (1200 UTC) – Hurricane Isidore weakens into a Category 2 hurricane.", "\nSeptember 29\n2:00 am AST (0600 UTC) – Hurricane Isidore weakens into a Category 1 hurricane.", "\n8:00 pm AST (0000 UTC September 30) – Hurricane Isidore weakens into a tropical storm.", "\n\nOctober\nOctober 1\n8:00 am AST (1200 UTC) – Tropical Storm Isidore weakens into a tropical depression.", "\n8:00 pm AST (0000 UTC October 2) – Tropical Depression Isidore becomes extratropical.", "\nOctober 4\n\n1:00 pm CDT (1800 UTC) – Tropical Depression Ten forms about 70 miles (110 km) east of the Tamaulipas/Veracruz border.", "\nOctober 6\n1:00 pm CDT (1800 UTC) – Tropical Depression Ten strengthens into Tropical Storm Josephine.", "\nOctober 7\n11:30 pm EDT (0330 UTC October 8) – Tropical Storm Josephine makes landfall in Taylor County, Florida, with 70 mph (110 km/h) winds.", "\nOctober 8\n2:00 am EDT (0600 UTC) – Tropical Storm Josephine becomes extratropical along the Georgia/Florida border.", "\nOctober 11\n8:00 am EDT (1200 UTC) – Tropical depression Eleven forms about midway between Belize and Great Swan Island.", "\n2:00 pm EDT (1800 UTC) – Tropical depression Eleven strengthens into Tropical Storm Kyle.", "\nOctober 12\n7:00 am CDT (1200 UTC) – Tropical Storm Kyle weakens into a tropical depression.", "\n1:00 pm CDT (1800 UTC) – Tropical Depression Kyle makes landfall near the Guatemala/Honduras border with 30 mph (50 km/h) winds.", "\n7:00 pm CDT (0000 UTC October 13) – Tropical Depression Kyle dissipates near where it made landfall.", "\nOctober 14\n\n8:00 am EDT (1200 UTC) – Tropical depression Twelve forms just east of Nicaragua.", "\nOctober 16\n2:00 am EDT (0600 UTC) – Tropical depression Twelve strengthens into Tropical Storm Lili.", "\nOctober 17\n8:00 am EDT (1200 UTC) – Tropical Storm Lili strengthens into Hurricane Lili.", "\nOctober 18\n5:30 am EDT (0930 UTC) – Hurricane Lili reaches Category 2 intensity and makes landfall on the Zapata Peninsula with 100 mph (160 km/h) winds.", "\n2:00 pm EDT (1800 UTC) – Hurricane Lili weakens into a Category 1 hurricane.", "\n8:00 pm EDT (0000 UTC October 19) – Hurricane Lili regains Category 2 intensity.", "\nOctober 19\n8:00 am EDT (1200 UTC) – Hurricane Lili reaches Category 3 intensity.", "\n2:00 pm EDT (1800 UTC) – Hurricane Lili weakens into a Category 2 hurricane.", "\n8:00 pm EDT (0000 UTC October 20) – Hurricane Lili weakens into a Category 1 hurricane.", "\nOctober 20\n\n8:00 am AST (1200 UTC) – Hurricane Lili regains Category 2 intensity.", "\n2:00 pm AST (1800 UTC) – Hurricane Lili weakens into a Category 1 hurricane.", "\nOctober 25\n8:00 am AST (1200 UTC) – Hurricane Lili regains Category 2 intensity.", "\n2:00 pm AST (1800 UTC) – Hurricane Lili weakens into a Category 1 hurricane.", "\nOctober 26\n2:00 pm AST (1800 UTC) – Hurricane Lili weakens into a tropical storm.", "\nOctober 27\n1:00 am AST (0600 UTC) – Tropical Storm Lili becomes extratropical about 300 nautical miles (345 miles, 550 km) north-northwest of the Azores.", "\n\nNovember\nNovember 16\n1:00 pm EST (1800 UTC) – Tropical depression Thirteen forms south of Jamaica.", "\nNovember 19\n1:00 am EST (0600 UTC) – Tropical depression Thirteen strengthens into Tropical Storm Marco.", "\nNovember 20\n\n1:00 am EST (0600 UTC) – Tropical Storm Marco strengthens into Hurricane Marco.", "\n1:00 pm EST (1800 UTC) – Hurricane Marco weakens into a tropical storm.", "\nNovember 23\n1:00 pm EST (1800 UTC) – Tropical Storm Marco weakens into a tropical depression.", "\nNovember 24\n7:00 am EST (1200 UTC) – Tropical Depression Marco regains tropical storm status.", "\nNovember 26\n7:00 am EST (1200 UTC) – Tropical Storm Marco weakens into a tropical depression.", "\n1:00 pm EST (1800 UTC) – Tropical Depression Marco dissipates about 100 miles (160 km) southwest of the Isle of Youth.", "\nNovember 30\n11:59 pm EST (0359 UTC) – The 1996 Atlantic hurricane season officially ends.", "\n\nSee also\n1996 Atlantic hurricane season\n\nNotes\n\nReferences\n\nCategory:1996 Atlantic hurricane season\n1996 Atlantic hurricane season\nCategory:Articles which contain graphical timelines" ]

{ "pile_set_name": "Wikipedia (en)" }

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[ "Edward Bartley\n\nEdward Bartley (23 February 1839 – 28 May 1919) was a British-born New Zealand architect. ", "Beginning as a builder, Bartley transitioned into a career as an architect, not an uncommon occurrence in the 19th century. ", "He is responsible for designing more than 20 churches and some of Auckland's most notable buildings.", "\n\nLife and career\n\nEarly life\nBartley was born to Robert Bartley (18 October 1798 – 9 April 1857) and Elizabeth ‘Betsy’ Benest (27 December 1801 – 24 March 1856) in Saint Helier, Jersey, Channel Islands.", "\nThe Bartley family lived in a tight-knit community in Saint Helier. ", "They also had family connections on the island of Guernsey, mainland Europe and in the United States. ", "The Bartley family resided in a home in Union Court, St Helier – a short lane running off the north side of Union Street. ", "The family home was designed and built by Edward Bartley's father, Robert Bartley. ", "\nEdward Bartley was the tenth child born to his parents. ", "Bartley's parents had twelve children, however, only five would survive through childhood.", "\nThe Bartley children were well educated according to standards of the time. ", "Bartley received lessons in drawing, drafting, music, calculation, languages, classics and mathematics. ", "Bartley also acquired a taste for science and technologies which were advancing rapidly at the time. ", "In particular, the advancements made in the development of the microscope, and in photography, were of interest to him all his life.", "\n\nImmigration to New Zealand and career beginnings\nThe Bartley children were well educated, according to the standards of the day. ", "Edward received a solid grounding in drawing, drafting, music and calculation, in addition to the languages, classics and mathematics. ", "He had also acquired a taste for the new science and technologies, which were developing swiftly at that time. ", "In particular the advancements made in the development of the microscope, and in photography, continued to interest him all his life.", "\n\nEdward Bartley began his career as an apprentice to his father, at the age of thirteen. ", "In this field, he followed his older brother, Robert, who had already qualified as a master builder. ", "\nAfter two years, Edward Bartley left Jersey having acquired some work experience and the sale-able skills of building and carpentry.", "\nBartley made the decision to leave Jersey due to the flat local economy which had resulted in no further public building projects. ", "\nBartley's father, Robert, was of an age to retire. ", "He had also contracted tuberculosis, a disease which had already claimed the lives of several family members.", "\nIn June 1854, Edward Bartley departed from London with his older brother Robert and his family with their sights set on New Zealand. ", "There were a number of destinations available to immigrate to, but the possibility of land grants to approved settlers may have tipped the balance in favour of the Bartley brothers choosing to immigrate to New Zealand. ", "The Auckland local government was keen to attract immigrants and had advertised in the British Isles the opportunity to receive free grants of land in the Auckland area.", "\nHaving made their plans to leave Jersey one year prior to their departure, the Bartley brothers would have felt their decision a timely one. ", "In March of that same year, conditions in Europe had begun to deteriorate, leading ultimately to Britain declaring war against the Russian Empire. ", "\nBartley accompanied his brother, Robert, to London – along with them followed Esther Kerby, wife of Robert Bartley, their young children and their possessions.", "\nThe Bartley brothers set sail for New Zealand on 20 June 1854 aboard the Joseph Fletcher. ", "After a passage of 115 days, the Bartleys reached New Zealand in early October.", "\nAuckland in 1854 was no sophisticated port settlement. ", "In the absence of a wharf, the ships anchored midstream. ", "Passengers and goods were transferred to lighters and run ashore on to the beach to disembark. ", "In some weathers this could be a hazardous and nerve-wracking end to a long sea journey, especially so for parents of young children who had been penned up for several months. ", "From the beach, the fortunate acquired drays to transport themselves and their possessions further into the town, the usual disembarking point being approximately the corner of the present day Queen and Shortland Streets.", "\n\nThere was an abundance of building work available in Auckland. ", "The primary obstacle in building then was the scarcity of materials. ", "Men with good training, who could easily adapt to colonial methods were desperately needed. ", "Men who were not above dressing their own timber in order to get the job done. ", "It was hard physical work that was waiting to be done and there was plenty of it.", "\n\nBartley himself recorded his reminiscences of his first experience of this new environment: \n\"We landed on a Thursday and commenced work on the following Monday, our first employer being A. Black, who was about to erect a building of five two-storey shops on the corner of Queen and Victoria Street East for J.S. McFarlane. ", "We started by placing into position wood blocks for the foundations, after which we proceeded with the wood framework, all of which work was so strange to us as joiners and so vastly different to the employment we had been accustomed to at home. \"", "There were no timber mills and therefore all the boards had to be hand planed and the tongue and groove worked by hand. ", "This was termed 'flogging the boards' and I found it anything but easy work to be constantly employed at from 6 o'clock in the morning until 6 in the evening\".", "\n\nBy 1857, Bartley was working with a Mr E.I. Matthews, a retired office of the Royal Engineers Department. ", "In that year, they began work on the Mount Eden Gaol, under the supervision of architect Reader Wood. ", "Bartley noted the extent of freedom allowed to the prisoners laboring on the site and their enterprise, particularly a shoemaker who undertook repairs to the worker's boots, for which payment was lodged with the authorities for collection at the completion of his term of detention.", "\n\nThe 1858 Militia Act\nThe 1858 Militia Act had divided the country into military districts and a system of ballot was introduced. ", "Allowances were made, however, for men to substitute service in volunteer companies in satisfaction of their militia duties and a great number did so, enabling a continuation of business and trade while satisfying their obligations to the nation's defence. ", "Edward became a member of the Royal Rifle Company of Volunteers. ", "Establishing himself in steady employment in Auckland as a builder, he married in February, 1859.", "\nThe following year was a tumultuous one, both at home and abroad. ", "The first battle of the Taranaki Wars at Waitara took place in March. ", "Tension peaked around the country. ", "By 1861 Governor Browne was still making heavy weather of the crisis in New Zealand. ", "It would be September before Governor Grey arrived in Auckland to take command. ", "By this time the American Civil War was underway and the growing international tensions made it difficult for the Governor to press his case for further resources.", "\nWhile all of this was well reported and debated in Auckland, building work continued steadily. ", "The escalation of war assured it. ", "Even the discovery of gold in Otago had not yet had an economic impact on the town, although wealth and influence were already moving inexorably southwards.", "\nIn 1862 Edward was now foreman for Mr Matthews and was engaged with his employer in demolishing the original St Paul's Anglican Church in Emily Place. ", "Edward recalled that Colonel Mould of the Royal Engineers was the architect and the design was highly regarded. ", "The colonel was a man of many talents. ", "Active in the Taranaki, he returned to Auckland and set about organising recruitment, as well as supervising the provision of more satisfactory roads, so essential for improving troop movement to the Waikato. ", "Edward now became Orderly-Sergeant of the No 5 Militia.", "\n\nWreck of Orpheus\nThe next year brought with it a great deal of tension and activity. ", "It was, however, events on the sea, not land, which brought trauma and sadness to the town. ", "Today it is difficult to appreciate the extent of the shock and grieving which affected Auckland when news came of the wreck of HMS Orpheus on 7 February. ", "A subscription was immediately taken up in the town for the relief of personal hardship occasioned by the tragedy and the newspapers of the day gave considerable coverage to the event and its aftermath.", "\nIn his memoirs Edward wrote:\n\"I remember the wreck of the HMS Orpheus 7 February 1863, which took place on the Manukau Bar. ", "The first we knew of the affair was by seeing drayloads of sailors being brought into Auckland. ", "Commodore Burnett and 189 officers and men were drowned and for days after the wreck bodies were being washed ashore. ", "Three officers succeeded in reaching the shore on a plank of teak from the wreck, and from this I made for them several mementoes such as picture frames, paper knives ... \"\nIt was over the winter of 1863 that a war mentality began to characterise the Auckland community. ", "Edward was ordered to the front that July, but his active service was short-lived. ", "Eleven tradesmen were required to return to complete the Fort Britomart stores, as capacity was fast being outstripped by demand.", "\nMr Matthews and Edward, now aged 26, entered into a partnership in 1865 as Matthews and Bartley Builders. ", "It was a difficult time for any new venture. ", "The capital had been removed from Auckland in February, with the attendant loss of personnel and government contracts.", "\nThe goldfields of the south continued to pull people away to those areas of the country with more promise of wealth and few \"native\" concerns. ", "The removal of the bulk of British troops was by now inevitable, no matter how much decried by the Auckland businessmen and speculators. ", "The partnership, however, seemed fortunate in regular contracts. ", "The Wesleyan Church in Pitt Street was one such, being completed and opened in October 1866. ", "The Supreme Court building was another.", "\n\nProfessional independence and further pursuits\nFebruary 1870 saw Edward moving to a greater degree of independence with the lease of offices in Albert Street, on his own behalf. ", "The Auckland Directory of 1873–4 shows that he had moved to Market Place by then, but still as a carpenter and builder. ", "On the evening of 9 April 1870 Edward laid aside commerce to pursue one of his cultural interests at the offices of Henry Partington in Queen Street. ", "The occasion was the inaugural meeting of the Society of Artists. ", "James Baker took the chair for the meeting, at which the following committee was proposed: J Baker, T Warner, T S Hall, T Symons, Dr F Wright, Mr Eastwood and E Bartley. ", "From then on meetings were regularly held, the next occasion using the YMCA premises as a venue, where an annual exhibition was proposed. ", "A sub-committee was formed to organise the forthcoming event, informally styled \"the Hanging Committee\". ", "The exhibition opened in February 1871. ", "Edward's involvement was to continue for a number of years, both in his capacity as office-holder and as an exhibitor.", "\n\nBy 1871 he had settled in the North Shore district of Devonport. ", "He built his family home, which still stands today, on the corner of Victoria and Calliope Roads. ", "Bartley Terrace, which runs below the house, was named for him. ", "By the early 1880s Edward was working as a designer and architect in his own right. ", "The Auckland Savings Bank building in Queen Street; the Abbott's Opera House; St John's Church Ponsonby; the Jewish synagogue in Princes Street; and the Holy Trinity church in Devonport are just some of his commissions in this period. ", "There was much in the Devonport district to attract his considerable energies, besides his career. ", "Edward shared with other men of his generation a sense of duty to his fellow citizens. ", "Such a philosophy was widely held, not just within his Masonic connections, but also as a virtue of the Victorian attitudes of service and charity to one's community. ", "He served on the Devonport Roads Board from its first meeting in 1883 and progressed from there to the first borough council of Devonport in 1886.", "\n\nFrom his earliest days in New Zealand Edward had been a supporter of the \"eight-hour day\" movement. ", "He was also fully in favour of free, secular and compulsory education. ", "He firmly believed it was the responsibility of the community to provide an opportunity for even the poorest to pass to highest qualification in his chosen trade or profession.", "\nAs a member of the Devonport Jubilee Committee he was instrumental in the opening of a new amenity in August 1887. ", "The Devonport Free Reading Room was opened \"to provide facilities for acquiring and disseminating literary and scientific knowledge\". ", "Edward was to demonstrate his microscope at this venue often, along with lectures and participation in various scientific societies and organisations held in the library or council chambers of an evening. ", "Edward served on the board of the Devonport School for 35 years and it was in a room of that school that he began his technical education classes for the boys of the district in 1891. ", "Many young men were to learn there the skills of joinery, carpentry and technical drawing. ", "In 1895 he broadened his educational interests to wider Auckland, as one of ten who attended a meeting on 2 October 1892 to establish the Auckland Technical School. ", "The first meeting of subscribers was held in 1895 and the school was opened in Rutland Street by Sir Maurice O'Rork on 10 June 1895. ", "This institution continues today as the Auckland University of Technology.", "\n\nThe Auckland Exhibition of 1898 was a marvellous occasion for the city. ", "Edward served on the building committee and experienced further satisfaction from the exhibits. ", "His technical school students collected first-class honours for a number of articles designed and constructed by them. ", "In January 1899, as part of the Exhibition, Edward participated in the \"Scientific Conversazione\" at the Choral Hall. ", "This evening of science and exploration gathered together over 100 microscopes and assorted instruments for the edification and enjoyment of the public. ", "This extraordinary display was organised by Edward, along with Professor Thomas and Mr Petrie. ", "Edward continued in his career as architect in the new century, serving as vice-president of the Institute of Architects in 1902. ", "In his later years, his sons carried on his contribution to Freemasonry, to the arts and music, and to community service. ", "Bartley's designs can be located all throughout the North Island of New Zealand and his influence extended to the next generation of New Zealand architects, notably Keith Draffin and Gerard Jones.", "\n\nHe worked mainly in Auckland. ", "Notable buildings include:\n Auckland Savings Bank Building, Queen Street, 1881\n St John's Church, Ponsonby Road, 1882\n Jewish synagogue, now University House, Princes Street, 1884\n Costley Home for the Aged Poor, Greenlane, 1889.", "\n Jubilee Building, Royal Institute for the Blind, Parnell, 1892\n Kings Theatre (now Mercury Theatre), Upper Pitt Street (now Mercury Lane), Newton, 1910\n Mt Eden Public Library, 1912.", "\n\nAll these buildings have Category One New Zealand Historic Places Trust designation\n\nHe trained his sons Alfred, Arthur and Frederick in his office. ", "The firm of Bartley & Son continued under Alfred Bartley following the retirement of Edward Bartley in 1914.", "\n\nFamily\nOn 16 February 1859, he married Elizabeth Hannken at the home of her father in Auckland. ", "She was a daughter of the German settler Frederick Hannken and his wife Eliza (daughter of Count and Countess De Granche). ", "The young couple lived first in Union Street Auckland, on part of the Hannken property. ", "They later moved to The Strand, North Shore at Devonport. ", "\nEdward's wife Elizabeth was born in Sydney, Australia in October 1838 and came to New Zealand in 1840 with her mother. ", "The Hannken family settled first in the Coromandel, later moving to Auckland. ", "Elizabeth met Edward through the Auckland Choral Society. ", "They were both amongst the foundation members in 1856.", "\n\nWith Elizabeth, he had 13 children. ", "Only 9 of their children would survive through til adulthood: \nArthur Edward (1859– 1940)\nFrederick Adolphous (1862– 1899)\nAlfred Martin (186?- 1929)\nMatilda Louisa (1867–1868)\nEmily Bertha (1869–1944)\nHarold Edgar (1871– 1872)\nMabel Theresa (1872–1873)\nAlbert Ernest (1873– 1940)\nMay Elizabeth (1875– 1951)\nEva Rosine (1877– 1954)\nPercival Leonard (1878– 1908)\nAmy Zealandia (1879– 1880)\nClaude Victor (1881– 1919)\nEdward was the great-uncle of Illustrator Harry Rountree, the son of his niece by his elder brother Robert, Julia Bartley.", "\n\nDeath\nBartley died at his home in Devonport on 28 May 1919, at the age of 80 and was buried at O'Neill's Point Cemetery. ", "His wife Elizabeth would die two years later on 27 December 1921.", "\n\nHis obituary appeared in the Taranaki Daily News on 31 May 1919: \"Mr. Edward Bartley, architect, aged 81 years, was found dead in bed on Wednesday. ", "The late Mr. Bartley was born in Jersey, Channel Islands, and was educated there. ", "He received his early training from his father, who was an architect and builder. ", "In 1854 he came to New Zealand in the ship Joseph Fletcher, in company with his elder brother, Mr. Robert Bartley. ", "For the past quarter of a century he had devoted his energies exclusively to architecture, designing and supervising the erection of several of Auckland's most handsome buildings, including the Auckland Savings Bank, the Jewish Synagogue, Opera House, and other noticeable structures. ", "Other noteworthy specimens of the late gentleman's skill and judgment are several city and suburban churches. ", "He was diocesan architect for the Church of England, and was widely experienced in church architecture. ", "The late Mr. Bartley was one of the first members of the Auckland Choral Society, established in 1855, and for fourteen years was choirmaster of the Trinity Church, Devonport, where he had resided for many years. ", "One of his many good works was the establishment of a boys' workshop (in connection with the district school), where for two nights a week for ten years he gave instruction .The workshop took a first award at the Industrial Exhibition in Auckland. ", "The late Mr. Bartley was a devoted lover of science, his scientific use of the microscope having been a hobby for over forty years.\"", "\n\nReferences\n\nCategory:New Zealand architects\nCategory:1839 births\nCategory:1919 deaths\nCategory:People from Saint Helier\nCategory:Jersey emigrants to New Zealand\nCategory:Burials at O'Neill's Point Cemetery" ]

{ "pile_set_name": "Wikipedia (en)" }

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[ "Ramona Strugariu\n\nRamona Strugariu is a Romanian politician currently serving as a Member of the European Parliament for the Freedom, Unity and Solidarity Party.", "\n\nIn addition to her committee assignments, Strugariu is a supporter of the European Parliament Intergroup on Anti-Corruption.", "\n\nReferences\n\nCategory:Living people\nCategory:MEPs for Romania 2019–2024\nCategory:Women MEPs for Romania\nCategory:21st-century Romanian politicians\nCategory:21st-century Romanian women politicians" ]

{ "pile_set_name": "Wikipedia (en)" }

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[ "Three-dimensional fuse deposition modeling of tissue-simulating phantom for biomedical optical imaging.", "\nBiomedical optical devices are widely used for clinical detection of various tissue anomalies. ", "However, optical measurements have limited accuracy and traceability, partially owing to the lack of effective calibration methods that simulate the actual tissue conditions. ", "To facilitate standardized calibration and performance evaluation of medical optical devices, we develop a three-dimensional fuse deposition modeling (FDM) technique for freeform fabrication of tissue-simulating phantoms. ", "The FDM system uses transparent gel wax as the base material, titanium dioxide (TiO2 ) powder as the scattering ingredient, and graphite powder as the absorption ingredient. ", "The ingredients are preheated, mixed, and deposited at the designated ratios layer-by-layer to simulate tissue structural and optical heterogeneities. ", "By printing the sections of human brain model based on magnetic resonance images, we demonstrate the capability for simulating tissue structural heterogeneities. ", "By measuring optical properties of multilayered phantoms and comparing with numerical simulation, we demonstrate the feasibility for simulating tissue optical properties. ", "By creating a rat head phantom with embedded vasculature, we demonstrate the potential for mimicking physiologic processes of a living system." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Visibility of epidermoid tumors on steady-state free precession images.", "\nTo determine whether steady-state free precession sequences improve the MR visibility of epidermoid tumors in comparison with spin-echo images. ", "Patients were four women and three men with epidermoid tumors in the subarachnoid spaces. ", "MR was performed with a 1.5-T superconductive unit. ", "For steady-state free precession imaging, three-dimensional Fourier transform fast imaging with steady-state free precession (FISP) images were used (20-40/7 [repetition time/echo time], flip angle of 25 degrees). ", "The visualization and contrast-to-noise ratio were compared in FISP images and spin-echo images. ", "In one case, the images of FISP and fast low-angle shot were obtained with variable repetition times and flip angles to evaluate the best pulse sequences for the visualization of epidermoid tumors. ", "The contrast-to-noise ratios between tumors and cerebrospinal fluid ranged from 7.9 to 17.5 (average was 12.9) on FISP images. ", "The average of contrast-to-noise ratios on T1, T2, and proton density-weighted spin-echo images were 1.6, 2.0, and 4.2, respectively. ", "Three-dimensional Fourier transform FISP images best showed central nervous system and demonstrated epidermoid tumors excellently. ", "Epidermoid tumors in the subarachnoid spaces were better demonstrated on steady-state free precession (three-dimensional Fourier transform FISP) than on conventional spin-echo images." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "---\nabstract: 'Let $(R,{\\mathfrak{m} },k)$ be a Noetherian local ring with maximal ideal ${\\mathfrak{m} }$ and residue field $k$. The linearity defect of a finitely generated $R$-module $M$, which is denoted ${\\operatorname{ld}}_R(M)$, is a numerical measure of how far $M$ is from having linear resolution. ", "We study the linearity defect of the residue field. ", "We give a positive answer to the question raised by Herzog and Iyengar of whether ${\\operatorname{ld}}_R(k)<\\infty$ implies ${\\operatorname{ld}}_R(k)=0$, in the case when ${\\mathfrak{m} }^4=0$.'\naddress: 'Rasoul Ahangari Maleki, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran'\nauthor:\n- Rasoul Ahangari Maleki\ntitle: Linearity defect of the residue field of short local rings\n---\n\n1 cm\n\nIntroduction and notation\n=========================\n\nThis paper is concerned with the notion of the linearity defect of the residue field of a commutative Noetherian local ring. ", "This invariant was introduced by Herzog and Iyengar [@H-i] and has been further studied by Iyengar and Römer [@I-T], Şega [@S] and Nguyen [@Hop]. ", "Let us recall the definition of the linearity defect. ", "Throughout this paper $(R,{\\mathfrak{m} },k)$ will denote a commutative Noetherian local ring with maximal ideal ${\\mathfrak{m} }$ and residue field $k$. Let $$\\mathbf{F}: \\cdots \\rightarrow F_{n}\\xrightarrow{\\partial_{n}}F_{n-1}\\xrightarrow{\\partial_{n-1}} \\cdots\\rightarrow F_{1}\\xrightarrow{\\partial_{1}} F_{0}\\rightarrow 0$$ be a minimal complex (i.e. $\\partial_i(F_i)\\subseteq {\\mathfrak{m} }F_{i-1}$ for all $i\\geq 0$) of finitely generated free $R$-modules. ", "Then the complex has a filtration $\\{\\mathfrak{F}^p \\mathbf{F}\\}_{p\\geq\n 0}$ with $(\\mathfrak{F}^p \\mathbf{F})_i={\\mathfrak{m} }^{p-i}F_i$ for all $p$ and $i$ where, by convention, ${\\mathfrak{m} }^j=R$ for all $j\\leq 0$. The associated graded complex with respect to this filtration is called *the linear part* of $ \\mathbf{F}$ and denoted by ${\\operatorname{lin}}^R(\\mathbf{F})$. Let $N$ be an $R$-module. ", "The notation $R^{{\\operatorname{\\mathsf g}}}$ will stand for the associated graded ring $\\oplus_{i\\geq 0}{\\mathfrak{m} }^i/{\\mathfrak{m} }^{i+1}$ and $N^{{\\operatorname{\\mathsf g}}}$ for the associated graded $R^{{\\operatorname{\\mathsf g}}}$-module $\\oplus_{i\\geq 0}{\\mathfrak{m} }^iN/{\\mathfrak{m} }^{i+1}N$. By construction, ${\\operatorname{lin}}^R(\\mathbf{F})$ is a graded complex of graded free $R^{{\\operatorname{\\mathsf g}}}$-modules and has the property that ${\\operatorname{lin}}^R_n(\\mathbf{F})=F_n^{{\\operatorname{\\mathsf g}}}(-n)$, for all $n$. For more information about this complex, we again refer to [@H-i] and [@I-T]. ", "Let $M$ be a finitely generated $R$-module. ", "The *linearity defect* of $M$ is defined to be the number $${\\operatorname{ld}}_{R}(M):=\\sup\\{i\\in\\mathbb{Z}| {\\operatorname{H}}_{i}({\\operatorname{lin}}^{R}(\\mathbf{F})) \\neq 0\\},$$ where $\\mathbf{F}$ is a minimal free resolution of $M$. By definition, ${\\operatorname{ld}}_R(M)$ can be infinite and ${\\operatorname{ld}}_R(M)\\leq d$ is finite if and only if $({\\operatorname{Syz}}_d(M))^{{\\operatorname{\\mathsf g}}}$ has a linear resolution over the standard graded algebra $R^{{\\operatorname{\\mathsf g}}}$, where ${\\operatorname{Syz}}_d(M)$ is the $d$th syzygy module of $M$. In particular, ${\\operatorname{ld}}_R(M)=0$ if and only if $M^{{\\operatorname{\\mathsf g}}}$ has a linear resolution over $R^{{\\operatorname{\\mathsf g}}}$ and then ${\\operatorname{lin}}^{R}(\\mathbf{F})$ is a minimal graded free resolution of $M^{{\\operatorname{\\mathsf g}}}$. The notion of the linearity defect can be defined, in the same manner, for graded modules over a standard graded algebra $A$ over a field $k$. In [@H-i], the authors proved that if ${\\operatorname{ld}}_A(k)<\\infty$, then ${\\operatorname{ld}}_A(k)=0$. Motivated by this known result in the graded case, the following natural question raised in [@H-i].", "\n\n\\[loc1\\] If ${\\operatorname{ld}}_R(k)< \\infty,$ does it follow that ${\\operatorname{ld}}_R(k)=0$?", "\n\nIf $R^{{\\operatorname{\\mathsf g}}}$ is Cohen-Macaulay, $\\c{S}$ega [@S] showed that the question has positive answer in the case that $R$ is a complete intersection. ", "Also, she gave an affirmative answer when ${\\mathfrak{m} }^3=0$. Another positive answer to the question is given by the author and Rossi [@AR] when $R$ is of homogeneous type, that is $\\dim_k{\\operatorname{Tor}}^R_i(k,k)=\\dim_k\n{\\operatorname{Tor}}^{R^{{\\operatorname{\\mathsf g}}}}_i(k,k)$ for all $i$.\\\nIn this paper we show that this problem has an affirmative answer when ${\\mathfrak{m} }^4=0$. The proof relies on the existence of a DG algebra structure of a minimal free resolution of residue field $k$.\n\nPreliminaries and the main result\n=================================\n\nŞega provided an interpretation of linearity defect in term of vanishing of special maps. ", "For each $n\\geq 0$ and $i\\geq0$ we consider the map $$\\upsilon^{n}_i(M):{\\operatorname{Tor}}_{i}^R(M,R/{\\mathfrak{m} }^{n+1})\\rightarrow\n{\\operatorname{Tor}}_{i}^R(M,R/{\\mathfrak{m} }^{n})$$ induced by the natural surjection $R/{\\mathfrak{m} }^{n+1}\\rightarrow R/{\\mathfrak{m} }^n$. For simplicity, we set $\\upsilon^{n}_i:=\\upsilon^{n}_i(k)$ when $M=k$.\n\n[@S Theorem 2.2]\\[S\\] Let $M$ be a finitely generated $R$-module and $d$ be an integer. ", "Then the following conditions are equivalent.", "\n\n1. ", " ${\\operatorname{ld}}_R(M)\\leq d;$\n\n2. ", " $\\upsilon^{n}_i(M)=0$ for all $i\\geq d+1$ and all $n\\geq 0$.\n\n\\[first\\] Let $i\\geq 0$. Assume that $\\mathbf{F}$ is a minimal free resolution of a finitely generated $R$-module $M$. Then by [@S 2.3 $(2^{'})$], the following statements are equivalent.", "\n\n1. ", " $\\upsilon^{1}_i(M)=0;$\n\n2. ", " if $x\\in F_{i}$ satisfies $\\partial_{i}(x)\\in {\\mathfrak{m} }^2F_{i-1}$, then $x\\in\n {\\mathfrak{m} }F_{i}$.\n\nLet $S$ be a unitary commutative ring. ", "Given an $S$-complex $\\mathbf{C}$, we write $|c|=i$ (the homological degree of $c$) when $c\\in\\mathrm{C}_i$. When we write $c\\in \\mathbf{C}$ we mean $c\\in C_i$ for some $i$. A (graded commutative) *DG algebra* over $S$ is a non-negative $S$-complex $(\\mathbf{D},\\partial)$ with a morphism of complexes called the product $$\\begin{aligned}\n\\mu^{\\mathbf{D}}: &\\mathbf{D}\\otimes_S\\mathbf{D}\\rightarrow \\mathbf{D}& \\\\\n&a \\otimes b\\mapsto ab&\\end{aligned}$$ satisfying the following properties:\n\n1. ", " unital: there is an element $1\\in\\mathrm{D}_0$ such that $1a=a1=a$ for $a\\in\\mathbf{D} $;\n\n2. ", " associative: $a(ba)=(ab)c$ for all $a,b,c\\in \\mathbf{D}$;\n\n3. ", " graded commutative: $ab=(-1)^{|a||b|}ba\\in D_{|a|+|b|}$ for all $a,b\\in \\mathbf{D}$ and $a^2=0$ when $|a|$ is odd.", "\n\nThe fact that $\\mu$ is a morphism of complexes is expressed by the *Leibniz rule*: $$\\partial(ab)=\\partial(a)b+(-1)^{|a|}a\\partial(b)$$ For more information on DG algebras we refer to [@Av].", "\n\n\\[homology\\] If $(\\mathbf{D},\\partial)$ is a DG algebra over $S$. Using Leibniz rule, one can see that the subcomplex of cycles $Z(\\mathbf{D})$ is a DG subalgebra of $\\mathbf{D}$ and the boundaries $B(\\mathbf{D})$ is a DG ideal of $Z(\\mathbf{D})$. Thus the product on $\\mathbf{D}$ induces a product on the homology $H(\\mathbf{D})=Z(\\mathbf{D})/B(\\mathbf{D})$. In particular, $\\oplus_{i\\geq 0} H_n(\\mathbf{D})$ is a graded module over commutative ring $H_0(\\mathbf{D})$.\n\nTate constructed a DG algebra (free) resolution of $k$. Furthermore, such a resolution can be chosen to be minimal, see [@Av Theorem 6.3.5], which we refer a minimal Tate resolution of $k$ over $R$.\\\nThe following lemma shows that the linear part of a minimal Tate resolution of $k$ inherits a DG algebra structure from that of the resolution.", "\n\n\\[dg\\] Let $(\\mathbf{F},\\partial)$ be a minimal Tate resolution of $k$. Then ${\\operatorname{lin}}^R(\\mathbf{F})$ has a DG algebra structure induced by that of $\\mathbf{F}$.\n\nLet $\\mu^{\\mathbf{F}}:\\mathbf{F}\\otimes_R\\mathbf{F}\\rightarrow\n\\mathbf{F}$ be a morphism of complexes which defines the product on $\\mathbf{F}$. Set $S=R^{{\\operatorname{\\mathsf g}}}$. Since $\\mathbf{F}$ is minimal we see that $\\mathbf{F}\\otimes_R\\mathbf{F}$ is a minimal complex as well. ", "Hence the morphism induces a morphism of graded $S$-complexes ${\\operatorname{lin}}^R(\\mu^{\\mathbf{F}}):\n{\\operatorname{lin}}^R(\\mathbf{F}\\otimes_R\\mathbf{F})\\rightarrow {\\operatorname{lin}}^R(\\mathbf{F})$ such that if $i,n\\geq 0$ and $x^*$ is the image of an element $x\\in\n{\\mathfrak{m} }^i(\\mathbf{F}\\otimes_R\\mathbf{F})_n $ in ${\\mathfrak{m} }^i(\\mathbf{F}\\otimes_R\\mathbf{F})_n/{\\mathfrak{m} }^{i+1}(\\mathbf{F}\\otimes_R\\mathbf{F})_n $, then ${\\operatorname{lin}}^R(\\mu^{\\mathbf{F}})$ maps $x^*$ into the image of $\\mu(x)$ in ${\\mathfrak{m} }^iF_n/{\\mathfrak{m} }^{i+1}F_{n}$. There is also a natural isomorphism of graded $S$-complexes $\\lambda:{\\operatorname{lin}}^R(\\mathbf{F})\\otimes_S {\\operatorname{lin}}^R(\\mathbf{F})\\rightarrow\n{\\operatorname{lin}}^R(\\mathbf{F}\\otimes_R\\mathbf{F})$ such that if $i,j,n,m\\geq 0$ and $x\\in {\\mathfrak{m} }^{i}F_n$ and $y\\in {\\mathfrak{m} }^{j}F_m$ with images $x^*$ in ${\\mathfrak{m} }^{i}F_n/{\\mathfrak{m} }^{i+1}F_n $ and $y^*$ in $ {\\mathfrak{m} }^{j}F_m/ {\\mathfrak{m} }^{j+1}F_m$ respectively, then $\\lambda$ maps $x^*\\otimes y^*$ into the image of $x\\otimes y$ in ${\\mathfrak{m} }^{i+j}(\\mathbf{F}\\otimes_R\\mathbf{F})_{n+m}/{\\mathfrak{m} }^{i+j+1}(\\mathbf{F}\\otimes_R\\mathbf{F})_{n+m}$ , see [@I-T Lemma 2.7]. ", "Now, define (the product) $$\\mu^{{\\operatorname{lin}}^R(\\mathbf{F})}: {\\operatorname{lin}}^R(\\mathbf{F})\\otimes_S {\\operatorname{lin}}^R(\\mathbf{F})\\rightarrow\n{\\operatorname{lin}}^R(\\mathbf{F})$$ as the composition ${\\operatorname{lin}}^R(\\mu^{\\mathbf{F}})\n\\circ\\lambda $. ", "Since $\\mu$ satisfies conditions $(i),(ii),(iii)$ of the definition of DG algebras, one can see that $\\mu^{{\\operatorname{lin}}^R(\\mathbf{F})}$ satisfies the same properties as well. ", "Therefore the linear part of $\\mathbf{F}$ is a DG algebra over $S$ augmented to $k$.\n\nLet ${\\mathfrak{m} }^{*}$ denote the homogeneous maximal ideal of $R^{{\\operatorname{\\mathsf g}}}$. The following is a direct consequence of the above lemma.", "\n\n\\[ann\\] If $\\mathbf{F}$ is a minimal free resolution of $k$, then ${\\mathfrak{m} }^{*}{\\operatorname{H}}_n({\\operatorname{lin}}^R(\\mathbf{F}))=0$ for all $n$.\n\nThe assertion follows from Remark \\[homology\\] with considering the fact that ${\\operatorname{H}}_0({\\operatorname{lin}}^R(\\mathbf{F}))=R^{{\\operatorname{\\mathsf g}}}/{\\mathfrak{m} }^{*}$.\n\nIn what will follow, let $(\\mathbf{F},\\partial)$ be a minimal free resolution of residue field $k$ with differential map $\\partial$. The differential map of ${\\operatorname{lin}}^R(\\mathbf{F})$ which is induced by $\\partial$ will be denoted by $\\partial^*$. We recall that ${\\operatorname{lin}}^R(\\mathbf{F})_n=F_n^{{\\operatorname{\\mathsf g}}}(-n)$. For any $ i,n\\geq 0$ and $x\n\\in {\\mathfrak{m} }^iF_n$, $\\partial^*$ maps $x+{\\mathfrak{m} }^{i+1}F_n$, the image of $x$ in ${\\mathfrak{m} }^iF_n/{\\mathfrak{m} }^{i+1}F_n$, into the image of $\\partial(x)$ in ${\\mathfrak{m} }^{i+1}F_{n-1}/{\\mathfrak{m} }^{i+2}F_{n-1}$ that is $\\partial(x)+\n{\\mathfrak{m} }^{i+2}F_{n-1}$.\n\n\\[one\\] Let $d$ be an integer. ", "If ${\\operatorname{ld}}_R(k)\\leq d$, then the following hold.", "\n\n1. ", " $\\upsilon^{1}_d=0$.\n\n2. ", " ${\\mathfrak{m} }^{*}{\\operatorname{Ker}}\\partial_{d}^{*}={\\mathfrak{m} }^{*} {\\operatorname{Im}}\\partial_{d+1}^{*}$.\n\nFor the simplicity, we set $\\mathrm{Z}={\\operatorname{Ker}}\\partial_{d}^{*}$ and $\\mathrm{B}= {\\operatorname{Im}}\\partial_{d+1}^{*}$.\\\n$(1)$ If $\\upsilon^{1}_d\\neq 0$, then there exists an element $e\\in F_{d}\\setminus {\\mathfrak{m} }F_{d}$ such that $\\partial_{d}(e)\\in\n{\\mathfrak{m} }^2F_{d-1}$, by \\[first\\] . ", "Let $e^*$ be the image of $e$ in the quotient module $F_d/{\\mathfrak{m} }F_d$. Then $\\partial_{d}^{*}(e^*)=0$ and so $e^*$ is a cycle in ${\\operatorname{lin}}^R(\\mathbf{F})$. Applying \\[ann\\], we have ${\\mathfrak{m} }^{*}\\mathrm{Z}\\subseteq \\mathrm{B}$ and therefore ${\\mathfrak{m} }^{*}e^*\\subseteq \\mathrm{B}$. As $e^*$ is an element of a basis of the free module $F^{{\\operatorname{\\mathsf g}}}_d(-d)$ and $\\mathrm{B}\\subseteq {\\mathfrak{m} }^{*}F^{{\\operatorname{\\mathsf g}}}_{d}(-d)$, it is straightforward to see that ${\\mathfrak{m} }^{*}e^*$ is a direct summand of $\\mathrm{B}$. By the hypothesis, $\\mathrm{B}$ has a linear resolution. ", "This implies that the same property holds for ${\\mathfrak{m} }^{*}e^*$. Therefore $k$ has a linear resolution over $R^{{\\operatorname{\\mathsf g}}}$ and consequently ${\\operatorname{lin}}^R(\\mathbf{F})$ is acyclic. ", "Hence $\\upsilon^{1}_d= 0$ and this is a contradiction.\\\nFor $(2)$, it is enough to show that ${\\mathfrak{m} }^{*}\\mathrm{Z}\\subseteq {\\mathfrak{m} }^{*}\\mathrm{B}$. First we claim that ${\\mathfrak{m} }^{*}\\mathrm{Z}$ is generated in degree at least $d+2$ and ${\\mathfrak{m} }^{*}\\mathrm{Z}\\subseteq \\mathrm{B}$. Indeed since $\\upsilon^{1}_d= 0$, applying Remark \\[first\\], one has $\\mathrm{Z}\\subseteq {\\mathfrak{m} }^{*}F^{{\\operatorname{\\mathsf g}}}_d(-d)$ and consequently ${\\mathfrak{m} }^{*}\\mathrm{Z}$ is generated in degree at least $d+2$. The second part of the claim follows from Corollary \\[ann\\].\\\nOn the other hand, $\\mathrm{B}$ has a linear resolution, by the hypothesis. ", "Hence $\\mathrm{B}$ is generated by elements of degree $d+1$ and then all its elements of degree at least $d+2$ contained in ${\\mathfrak{m} }^{*}\\mathrm{B}$. Now, putting these two considerations together, we get ${\\mathfrak{m} }^{*}\\mathrm{Z}\\subseteq {\\mathfrak{m} }^{*}\\mathrm{B}$.\n\nNow, we are ready to prove our main result.", "\n\nAssume that $R$ is Artinian with ${\\mathfrak{m} }^4=0$. If ${\\operatorname{ld}}_R(k)<\\infty$, then ${\\operatorname{ld}}_R(k)=0.$\n\nLet $d$ be a non-negative integer and ${\\operatorname{ld}}_R(k)\\leq d$. We prove by descending induction on $d$. The case where $d=0$ is clear. ", "Let $d>0$. Applying Proposition \\[one\\], we have $\\upsilon^{1}_d=0$. Since ${\\mathfrak{m} }^4=0$, it follows from [@S Theorem 7.1] that $\\upsilon^{2}_d=0$. Again since ${\\mathfrak{m} }^4=0$ obviously $\\upsilon^{i}_d=0$ for all $i\\geq 3$, by the definition of the map $\\upsilon^{i}_d$. Therefore, from \\[S\\] we get ${\\operatorname{ld}}_R(k)\\leq\nd-1$. This completes the induction and finishes the proof.", "\n\n1 cm **Acknowledgments.** ", "The author would like to express great thanks to the referee for valuable comments and suggestions which have improved the exposition of this paper. ", "This research was in part supported by a grant from IPM (No. ", "94130028). ", "This work was also jointly supported by the Iran National Science Foundation (INSF) and Alzahra University grant No. ", "95001343.", "\n\n[2]{}\n\nR. Ahangari Maleki, M.E. Rossi, Regularity and linearity defect of modules over local rings, J. Commut. ", "Algebra 6 (2014) 485-504.", "\n\nL.L. Avramov, Infinite free resolutions, in: Six Lectures on Commutative Algebra, Bellaterra, 1996, in: Progr. ", "Math., ", "vol. ", "166, Birkh$\\ddot{a}$user, Basel, 1998, pp. ", "1-–118.", "\n\nJ. Herzog, S. Iyengar, Koszul modules, J. Pure Appl. ", "Algebra 201 (2005) 154–188.", "\n\nS. Iyengar, T. Römer, Linearity defects of modules over commutative rings, J. Algebra 322 (2009) 3212-–3237.", "\n\nH.D. Nguyen, Notes on the linearity defect and applications, arXiv:1411.0261.", "\n\nL.M. Şega, On the linearity defect of the residue field, J. Algebra 384 (2013) 276–290.", "\n" ]

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[ "Comparison of stool microbiota compositions, stool alpha1-antitrypsin and calprotectin concentrations, and diarrhoeal morbidity of Indonesian infants fed breast milk or probiotic/prebiotic-supplemented formula.", "\nThe composition of faecal microbiota of babies is known to be influenced by diet. ", "Faecal calprotectin and α1-antitrypsin concentrations may be associated with mucosal permeability and inflammation. ", "We aimed to assess whether there was any difference after consumption of a probiotic/prebiotic formula on faecal microbiota composition, calprotectin and α1-antitrypsin levels, and diarrhoea in comparison with breast milk-fed Indonesian infants. ", "One hundred sixty infants, 2 to 6 weeks old, were recruited to the study. ", "They were either breastfed or formula fed (80 per group). ", "Faecal samples were collected at recruitment and 3 months later. ", "Bacterial groups characteristic of the human faecal microbiota were quantified in faeces by quantitative polymerase chain reaction. ", "Calprotectin and α1-antitrypsin concentrations were measured using commercial kits. ", "Details of diarrhoeal morbidity were documented and rated for severity. ", "The compositions of the faecal microbiota of formula-fed compared with breast milk-fed children were similar except that the probiotic strain Bifidobacterium animalis subsp. ", "lactis DR10 was more abundant after 3 months consumption of the formula. ", "Alpha1-antitrypsin levels were higher in breastfed compared with formula-fed infants. ", "The occurrence of diarrhoea did not differ between the groups of babies. ", "Feeding Indonesian babies with a probiotic/prebiotic formula did not produce marked differences in the composition of the faecal microbiota in comparison with breast milk. ", "Detrimental effects of formula feeding on biomarkers of mucosal health were not observed." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Q:\n\nWhat can I do to solve the moisture problem in my bathroom?", "\n\nMy bathroom walls are done in concrete not drywall. ", "I installed a ceiling fan. ", "After showering, the walls are covered with moisture. ", "\nWhat can I do to eliminate the moisture?", "\n\nA:\n\nCondensation happens whenever hot air meets cold surfaces, and the air after a shower is hot and very humid. ", "\nYou'll always get some condensation, but there are a few things done to mitigate this:\n\nNormal interior walls and well-insulated exterior walls are usually not a lot colder than the air temperature. (", "The colder the walls are, the more condensation will happen.)", "\nAn exhaust fan will remove some of the humid, hot air (and since hot air rises, the exhaust fan is on or near the ceiling which has the greatest effect), and regular (room temperature, and regular humidity) air will be drawn in from HVAC vents or under the bottom of the door. ", "Opening the door will help even more. ", "\nA semi-gloss or high-gloss paint is used (usually these are labelled Kitchen and Bath) which holds up better to high humidity, condensation and regular cleaning. ", "\n\nIf your walls are actually solid concrete or concrete blocks, then they definitely are going to be colder as concrete is a poor insulator. ", "This is why in the same room concrete will feel cooler to the touch than drywall, which will feel cooler to the touch than a towel. ", "\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "/**\n * Copyright (c) 2014-present, The osquery authors\n *\n * This source code is licensed as defined by the LICENSE file found in the\n * root directory of this source tree.", "\n *\n * SPDX-License-Identifier: (Apache-2.0 OR GPL-2.0-only)\n */\n\n#include <boost/optional.hpp>\n#include <unordered_map>\n\n#include <osquery/core/core.h>\n#include <osquery/filesystem/filesystem.h>\n\nnamespace osquery {\n// Information about a single mounted filesystem\nstruct MountInformation final {\n struct StatFsInfo final {\n // Optimal transfer block size (statfs::f_bsize)\n std::uint32_t block_size{0U};\n\n // Total data blocks in file system (statfs::f_blocks)\n std::uint32_t block_count{0U};\n\n // Free blocks in filesystem (statfs::f_bfree)\n std::uint32_t free_block_count{0U};\n\n // Free blocks available to unprivileged user (statfs::f_bavail)\n std::uint32_t unprivileged_free_block_count{0U};\n\n // Total file nodes in filesystem (statfs::f_files)\n std::uint32_t inode_count{0U};\n\n // Free file nodes in filesystem (statfs::f_ffree)\n std::uint32_t free_inode_count{0U};\n };\n\n // Filesystem type\n std::string type;\n\n // Device path\n std::string device;\n\n // Canonicalized device path\n std::string device_alias;\n\n // Mount path\n std::string path;\n\n // Mount options\n std::string flags;\n\n // statfs information; may not be set if the statfs operation\n // has failed\n boost::optional<StatFsInfo> optional_statfs_info;\n};\n\n// Information about all mounted filesystems\nusing MountedFilesystemMap = std::unordered_map<std::string, MountInformation>;\n\nStatus getMountedFilesystemMap(MountedFilesystemMap& mounted_fs_info);\n} // namespace osquery\n" ]

{ "pile_set_name": "Github" }

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[ "Emanuele Massai\n\nTravel Guide to Bangkok 99traveltips.com The Facts & Figures of Bangkok Photo credit With a population of around eleven million people, Bangkok is located in the centre of Thailand and is split into fifty districts, each with a style and personality of its own. ", "There are areas of Bangkok...\n\n10 Beautiful Beaches You Have To Visit In Thailand - Hand Luggage Only - Travel, Food & Home Blog" ]

{ "pile_set_name": "Pile-CC" }

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[ "GTribe, the world's most advanced network for tech enthusiasts and PC gamers, has partnered with DEEPCOOL, XFX, Radeon, Seasonic, GIGABYTE, G.SKILL, Logitech G and STEIGER DYNAMICS for an epic giveaway experience!", "\n\nONE (1) Grand Prize Winner will receive a super-powered gaming PC, handcrafted in Silicon Valley by STEIGER DYNAMICS. ", "This amazing PC is powered by Intel's 8th Gen Core i7-8700K 6-Core processor and an XFX Radeon RX Vega 64 8GB BP 3xDP GPU. ", "This out-of-the-world gaming PC is augmented by the latest Logitech G gear.", "\n\n\n\nThe Grand Prize value is $4,000 ." ]

{ "pile_set_name": "OpenWebText2" }

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[ "An interactive agency serving our customers by helping them leverage technology to grow their business\n\nDec 21, 2012\n\nNetSource 2nd Annual Bake-off\n\nWe don’t just make website at NetSource, our talented team members have also been known to make a tasty treat or two. ", "This year the competition will be steep, with lots of entries competing for the following prizes:\n\nTastiest\n\nMost Festive\n\nMost Artistic\n\nVoting is open to all NetSource employees (be sure to taste-test vigorously today!), ", "just add your comment below with your choice for each of the three categories." ]

{ "pile_set_name": "Pile-CC" }

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[ "A large study of Caucasian children has identified this SNP in the TBX21 gene (and one other, rs16947078) as defining a haplotype associated with increased risk for allergic asthma. [", "PMID 17949803]\n\nThe odds ratio associated with rs11650354(T;T) homozygotes compared to rs11650354(C;C) homozygotes for allergic asthma is reported to be 8.13 (CI: 2.5-26.9). ", "Heterozygotes may be at slightly higher risk. [", "PMID 17949803]\n\n[PMID 19950257] Polymorphisms in TBX21 and STAT4 increase the risk of systemic sclerosis: Evidence of possible gene-gene interaction and alterations in Th1/Th2 cytokines\n\n[PMID 18504423] Polymorphisms in inflammation-related genes are associated with susceptibility to major depression and antidepressant response." ]

{ "pile_set_name": "Pile-CC" }

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[ "As electronic devices become lighter, thinner, shorter, and smaller, an important factor in achieving such reductions is high density mounting of semiconductor packages, as key components of such electronic devices.", "\nIn a computer or other electronic device, sizes of semiconductor devices, such as a random access memory (RAM) and a flash memory, increase to achieve an increase in memory capacity, while packages are miniaturized. ", "Approaches for reducing package size have been developed. ", "For example, according to one approach, a stack-type semiconductor package is configured to include a plurality of stacked semiconductor chips or semiconductor device packages. ", "According to another approach, a semiconductor module having a plurality of semiconductor chips, a plurality of semiconductor device packages, and/or stack-type semiconductor packages are mounted on at least one surface of a printed circuit board (PCB).", "\nSuch packages may be classified into a semiconductor chip stack-type package, which is identical to a multi-chip package (MCP) including a plurality of semiconductor chips having different functions. ", "The semiconductor chip stack-type package includes a plurality of stacked semiconductor chips to realize a high capacity.", "\nThe semiconductor chip stack-type package may be a product or application of a three-dimensional (3D) arrangement technique. ", "With the 3D arrangement technique, a plurality of memory chips is stacked to improve the integrity of a semiconductor memory. ", "As a semiconductor chip stack-type package, a multi-channel DRAM device is developed to satisfy a need for high-density and high-capacity memories and to improve system performance.", "\nSince a multi-channel semiconductor memory device has a structure wherein a plurality of semiconductor chips (or channel memories) is stacked, through-silicon vias (TSVs) may be applied to the multi-channel semiconductor memory device. ", "A through-silicon via may be used as conductive lines between a plurality of memories. ", "The through-silicon via may function as a signal transfer line for transferring a command signal or an address signal applied from a memory controller or a line for transferring data.", "\nIn such channel memories, a plurality of vias is formed at semiconductor chips to penetrate a die vertically.", "\nIn a conventional multi-channel DRAM device, since a command signal and an address signal are applied to each memory device independently, each channel coupled with each memory necessitates dedicated command and address transfer lines. ", "Accordingly, a die area occupied by through-silicon vias is relatively large. ", "For example, if a command/address line number per channel is 23 and a channel number is 16, one die may require 368 through-silicon vias.", "\nAn increase in the number of through-silicon vias causes an increase in chip die size. ", "As a result, a yield of a multi-channel semiconductor memory device is lowered and cost is increased." ]

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[ "Q:\n\nUPDATE FROM another table value with CASE SUBSTRING and WHERE\n\nI tried to update my main table --> budget with values from item_master table with condition\nHere's the table structure \nbudget :\n+-------------------+---------------+------+-----+------------+----------------+\n| Field | Type | Null | Key | Default | Extra |\n+-------------------+---------------+------+-----+------------+----------------+\n| id | int(11) | NO | PRI | NULL | auto_increment |\n| department | varchar(50) | YES | | NULL | |\n| section | varchar(50) | YES | | NULL | |\n| budget_id | varchar(50) | NO | | NULL | |\n| carmaker | varchar(255) | YES | | NULL | |\n| carline | varchar(255) | YES | | NULL | |\n| phase | varchar(255) | YES | | NULL | |\n| purpose | varchar(50) | YES | | NULL | |\n| order_plan | varchar(50) | YES | | NULL | |\n| required_date | varchar(50) | YES | | NULL | |\n| subgroup | varchar(255) | YES | | NULL | |\n| item_desc | varchar(255) | YES | | NULL | |\n| item_code | varchar(255) | YES | | NULL | |\n| qty | int(11) | YES | | NULL | |\n| curr | char(3) | YES | | NULL | |\n| price | decimal(20,2) | YES | | NULL | |\n| amount_ori | decimal(20,2) | YES | | NULL | |\n| amount | decimal(20,2) | YES | | NULL | |\n| amount_reduce_ori | decimal(20,2) | NO | | NULL | |\n| amount_reduce_usd | decimal(20,2) | NO | | NULL | |\n| qty_reduce | decimal(20,2) | NO | | NULL | |\n| qty_final | int(11) | YES | | NULL | |\n| amount_final | decimal(20,2) | YES | | NULL | |\n| status | varchar(50) | YES | | uncommited | |\n| source | varchar(50) | YES | | NULL | |\n| coa_production | char(7) | YES | | NULL | |\n| coa_general | char(7) | YES | | NULL | |\n| hfm_cr | char(6) | YES | | NULL | |\n| hfm_pl | char(6) | YES | | NULL | |\n| update_record | varchar(50) | YES | | NULL | |\n| user | varchar(50) | YES | | NULL | |\n+-------------------+---------------+------+-----+------------+----------------+\n\nitem_master :\n+----------------+---------------+------+-----+---------+----------------+\n| Field | Type | Null | Key | Default | Extra |\n+----------------+---------------+------+-----+---------+----------------+\n| id | int(11) | NO | PRI | NULL | auto_increment |\n| subgroup | varchar(255) | NO | | NULL | |\n| item_desc | varchar(255) | NO | | NULL | |\n| item_code | varchar(255) | NO | | NULL | |\n| uom | varchar(255) | NO | | NULL | |\n| price | decimal(20,2) | NO | | NULL | |\n| coa_production | char(7) | NO | | NULL | |\n| coa_general | char(7) | NO | | NULL | |\n| hfm_cr | varchar(50) | NO | | NULL | |\n| hfm_pl | varchar(50) | NO | | NULL | |\n+----------------+---------------+------+-----+---------+----------------+\n\nI'm trying to get coa_production or coa general , depend on purpose field\nif the first three (3) characters = 'PRO' then use coa_production, else use coa_general with item_code as the unique string/foreign keys\nHere's my current code :\nUPDATE budget CASE\nWHEN SUBSTR(purpose,1,3) = 'PRO' THEN\nSET coa_production = (SELECT coa_production FROM item_master WHERE budget.item_code = item_master.item_code) ELSE\nSET coa_general = (SELECT coa_general FROM item_master WHERE budget.item_code = item_master.item_code) END\n\nmy goal is when purpose = 'PRO' then the only field that has to be filled is coa_production using the value from item_master table, so does if purpose = 'GEN' then the only field that has to be filled is coa_general\n\nA:\n\nI would express this using an update join:\nUPDATE budget b\nLEFT JOIN item_master im\n ON b.item_code = im.item_code\nSET\n b.coa_production = CASE WHEN LEFT(b.purpose, 3) = 'PRO' THEN im.coa_production END,\n b.coa_general = CASE WHEN LEFT(b.purpose, 3) = 'GEN' THEN im.coa_general END\nWHERE\n LEFT(b.purpose, 3) IN ('PRO', 'GEN');\n\nNote that a left join (rather than an inner join) is required here, to keep the behavior the same as your original query. ", " Using a left join, a NULL would be assigned to a column which did not find any matching record in the item_master table.", "\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "According to the players, the women were promotional models hired from Sirens Agency - which supplies \"strippers, lingerie and topless waitresses\".", "\n\nThey said a businessman friend, who booked the cruise, invited the girls.", "\n\n\"We have spoken to the players involved and are entirely comfortable that this was an innocent afternoon spent on a friend's boat in Sydney Harbour,\" The Mirror quoted a Rugby Football Union (RFU) spokesman, as saying.", "\n\nHe added: \"We are confident their behaviour was not in any way inappropriate.\"", "\n\nThe team were celebrating a 15-9 win over the Australian Barbarians - with Armitage, Haskell and Doran-Jones all starring. (", "ANI)" ]

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[ "Warning! ", "JavaScript deactivated!", "\n\nJava Script is deactivated or not supported by your Browser\n\nAbout Microsoft System Center Application Virtualization\n\nApplication Virtualization\n\nMicrsoft Applivation Virtualization (or \"App-V\") can be\nconsidered as one of the first Application Virtualization products\nfor Windows Operating Systems. ", "Application Virtualization isolates\nprograms, that are running on a Windows OS, from other applications\nand from the OS itself. ", "This essentially is done by\nredirecting (\"virtualizing\") function calls to the OS' File\nSystem, Registry and other objects. ", "Because System Calls are\nisolated, Application Virtualization allows to run\nmultiple Versions of one application at the same time or to\nrun applications, that usualy woulöd have file or registry\nconflicts, at the same time. ", "Also, the underlying\nOperating System is protected, because Application Virtualization\nalso may redirect modification attempts of System Components like\nWindows DLLs, cruicial Registry Keys and Values and others.", "\n\nBesides this mechanism of Application Isolation, AppVirt ofthen\nis refered to as \"Application Streaming\". ", "This, however, is not\npart of every Application Virtualization Product. ", "In\nApplication Virtualization Solutions, usually the Virtual\nApplcation Packge has to be downloaded to the Client, before it can\nbe executed (whereas the Client can be a Windows Workstation or\nTerminal/Citrix Server). ", "To decrease deployment time and network\ntraffic when deploying new Packages, Streaming comes up. ", "Streaming\nin that context means, that not the whole package is downloaded to\nthe client. ", "Rather, only a portion of the whole package (usually\n20-40%) are transfered from a central server to the clients. ", "This\nallows to start new applications quite fast, because they already\nrun shortly after this inital portion has been transferred.", "\n\nMicrosoft App-V is the successor of Softricity's SoftGrid,\n. ", "App-V's products cover both, Application Isolation and\nApplication Streaming feature. ", "In fact it was SoftGrid that\nestablished the terms \"Application Virtualization\" and \"Application\nStreaming\" to be synonyms.", "\n\nMicrosoft Application Virtualization History\n\nMicrosoft App-V is the successor of SoftGrid, a product that\noriginally was created by the Boston based Company Softricity\n\nMicrosoft bought Softricity in July 2006 (however it was almost\nclear that this acquisition will happen some months earlier).", "\nMicrosoft then released \"SoftGrid 4.1\" that was the first version\nwhich passed Microsoft's Quality Assurance Process. ", "Almost at the\nsame time Microsoft released SoftGrid 4.2 that mainly added Support\nfor Microsoft Vista.", "\n\nThe current version is App-V 4.5, that mainly adds the concept\nof \"Dynamic Suite Composition\" and a broad range of supported\nDeployment Technologies.", "\n\nApp-V 4.6, that will add 64-bit support for the Virtualization\nClient is available since February 2010 and was updated to Service\nPack 1 in February 2012" ]

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[ "#import \"WeatherCloudPersistenceDelegate-Protocol.h\"\n#import \"SynchronizedDefaultsDelegate-Protocol.h\"\n#import \"WeatherPreferencesPersistence-Protocol.h\"\n\n@class WeatherPreferences;\n\n@interface WeatherCloudPreferences : NSObject <WeatherCloudPersistenceDelegate> {\n\n\tid<SynchronizedDefaultsDelegate> _syncDelegate;\n\tid<WeatherPreferencesPersistence> _cloudStore;\n\tWeatherPreferences* _localPreferences;\n\n}\n\n@property (retain) id<WeatherPreferencesPersistence> cloudStore; //@synthesize cloudStore=_cloudStore - In the implementation block\n@property (retain) WeatherPreferences * localPreferences; //@synthesize localPreferences=_localPreferences - In the implementation block\n@property (assign,nonatomic) id<SynchronizedDefaultsDelegate> syncDelegate; //@synthesize syncDelegate=_syncDelegate - In the implementation block\n@property (readonly) unsigned long long hash; \n@property (readonly) Class superclass; \n@property (copy,readonly) NSString * description; \n@property (copy,readonly) NSString * debugDescription; \n-(void)setSyncDelegate:(id<SynchronizedDefaultsDelegate>)arg1 ;\n-(id)initWithLocalPreferences:(id)arg1 persistence:(id)arg2 ;\n-(void)setLocalPreferences:(WeatherPreferences *)arg1 ;\n-(void)setCloudStore:(id<WeatherPreferencesPersistence>)arg1 ;\n-(void)cloudCitiesChangedExternally:(id)arg1 ;\n-(id<WeatherPreferencesPersistence>)cloudStore;\n-(BOOL)legacyCloudCity:(id)arg1 isEqualToALCity:(id)arg2 ;\n-(id)cloudCityFromALCity:(id)arg1 name:(id)arg2 ;\n-(id)cloudCitiesFromLegacyCloudCities:(id)arg1 ;\n-(WeatherPreferences *)localPreferences;\n-(id)cloudCityRepresentationsFromLegacyRepresentations;\n-(id)reconcileCloudCities:(id)arg1 withLocal:(id)arg2 isInitialSync:(BOOL)arg3 ;\n-(BOOL)areCloudCities:(id)arg1 equalToLocalCities:(id)arg2 ;\n-(void)updateLocalStoreWithRemoteChanges:(id)arg1 ;\n-(void)saveCitiesToCloud:(id)arg1 ;\n-(id<SynchronizedDefaultsDelegate>)syncDelegate;\n-(id)citiesByEnforcingSizeLimitOnResults:(id)arg1 ;\n-(id)prepareLocalCitiesForReconciliation:(id)arg1 isInitialSync:(BOOL)arg2 ;\n-(id)cloudRepresentationFromCities:(id)arg1 ;\n-(BOOL)shouldWriteCitiesToCloud:(id)arg1 ;\n-(void)setCloudStoreCities:(id)arg1 ;\n-(void)cloudPersistenceDidSynchronize:(id)arg1 ;\n-(id)initWithLocalPreferences:(id)arg1 ;\n@end\n" ]

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[ "Meet Alabama Realtor Leslie Hopper\n\nLeslie is a lifelong resident of Etowah County and is a graduate of Gadsden High School. ", "She has been a Realtor since leaving Jacksonville State University in 1996. ", "After initial training and professional development, she began working for Realty Plus, Inc and consistently achieved professional honors, including sales awards. ", "After taking time away from the real estate profession to raise her two children, Hayden and Hudson, she is excited to turn her energy toward assisting clients in their home transitions.", "\n\nThere aren’t many people with a level of integrity like Leslie. ", "She leads her life with support from her husband Scott, and they both rely on prayer to direct their path. ", "She has an infectious personality and an inner strength that gives everyone a sense of peace.", "\n\nLet her work for you and help you discover your American Dream! ", "Give Leslie a call at Realty Plus, Inc if you are looking to buy or sell your home! ", "She will put her years of experience and expertise to work for you!", "\n\nA question I’m often asked is “Can I buy a house with no down payment?” ", "The answer is YES, it is possible! ", "Call me... I will help you find a home in an eligible area! ", "Leslie Hopper 256-312-3780 Realty Plus, Inc. Please click here to view \"No Down Payment\" (100% financing) qualified homes." ]

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[ "Management pitfalls in the use of embolization for the treatment of severe epistaxis.", "\nAngiographic embolization for the treatment of severe recurrent epistaxis was added to the traditional treatment options--nasal packing, cauterization, and surgical vessel ligation--in 1974. ", "Since then, clinical experience has shown that this procedure is safe and effective. ", "When epistaxis cannot be controlled with cautery, nasal packing is the most common next step. ", "As such, it is often performed by emergency physicians and other clinicians who are not otolaryngologists. ", "We report two cases in which intranasal neoplasms were obscured as a result of a significant distortion of the normal anatomy. ", "This distortion was secondary to emergency-room treatment of severe epistaxis by repeated nasal packing followed by angiographic embolization. ", "Pre-embolization angiographic studies and subsequent postembolization endoscopic evaluations did not reveal the presence of the occult neoplasms because of the presence of inflammation and edema after treatment. ", "Clinicians should be aware that nasal packing and embolization can obscure the underlying source of epistaxis, and follow-up radiologic studies and endoscopic evaluations are essential to avoid delays in diagnosis." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "9,876 Megabytes of private Blackbytes, click to claim them!", "\n\n\n\n(November 8, 2018)\n\nI don't trust the file\n\nUpload the file to VirusTotal to inspect with over 70 antivirus scanners.", "\n\nWhat are Blackbytes?", "\n\nOne of the two Byteball currencies. ", "Blackbytes are a cash-like untraceable currency whose transactions are not visible on the public database. ", "They are sent peer-to-peer instead.", "\n\nCan I send Blackbytes if I don't know the recipients wallet address?", "\n\nYes! ", "You can send them via email, WhatsApp, Telegram etc. ", "Read all about it here.", "\n\nCan I send Zcash or Monero without knowing the recipients wallet address?", "\n\nNo.", "\n\nWhere can I trade Blackbytes?", "\n\nThe most popular exchange is Freebe.", "\n\n\n\nWhy are Blackbytes not listed on Binance?", "\n\nIf Blackbytes were traded on a centralized exchange, the exchange would have enough information to reveal the origin and destination of Blackbytes, and amounts traded, which defeats their purpose.", "\n\nBlackbytes can only be traded on decentralized exchanges, with conditional payments in the Byteball wallet. ", "Peer discovery has to happen outside the Byteball wallet. ", "Such an exchange is yet (November 2018) to appear." ]

{ "pile_set_name": "OpenWebText2" }

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[ "The station was a stopover for Immobilizer cruisers that operated in the Outer Rim Territories.[2] The drydock was the main target of Operation Ram's Head. ", "In this operation, four Star Destroyers docked to the station were crippled by an automated CR90 corvette, the Ram's Head, sent on a ramming trajectory by the Rebel Alliance. ", "The corvette was fitted with the Shield X technology which rendered the vessel nearly invincible to damage from the attack. ", "The drydock itself was undamaged, but the attack was a humiliating setback for the Empire.[1]" ]

{ "pile_set_name": "Pile-CC" }

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[ "Assessment of treatment of Osteochondrosis dissecans (OCD) of shoulder joint in dogs--the results of two years of experience.", "\nOsteochondrosis is a common and clinically important joint disorder that occurs in human and many species of animals such as pigs, horses and dogs. ", "The aim of this article is to present the results of conservative and surgical treatment of Osteochondrosis dissecans (OCD) and to compare the recovery time in relation to race, age and extent of cartilage damage. ", "The study was performed on 36 dogs of both sexes, aged from 4 to 24 months, weighing 12-70 kg, of different breeds (predisposed and non-specific for this disease). ", "All patients underwent clinical, orthopedic and radiological examination through which the location of OCD changes in the shoulder joint and the degree of this problem were established. ", "Based on the examinations the patients were qualified for particular treatment procedures. ", "Control tests were performed after 14, 30 and 60 days, which allowed for a precise evaluation of the progress of therapy. ", "Achieved results indicate that dogs age and the lesions extent in the shoulder joint have an impact on the outcome of treatment. ", "Studies showed that the best therapeutic effects were obtained in dogs aged 6-10 months, which underwent surgery. ", "The study proved that the early diagnosis of OCD strongly influence the recovery rate and may inhibit the development of degenerative changes in the joint (DJD)." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "RBM28, a protein deficient in ANE syndrome, regulates hair follicle growth via miR-203 and p63.", "\nAlopecia-neurological defects-endocrinopathy (ANE) syndrome is a rare inherited hair disorder, which was shown to result from decreased expression of the RNA-binding motif protein 28 (RBM28). ", "In this study, we attempted to delineate the role of RBM28 in hair biology. ", "First, we sought to obtain evidence for the direct involvement of RBM28 in hair growth. ", "When RBM28 was downregulated in human hair follicle (HF) organ cultures, we observed catagen induction and HF growth arrest, indicating that RBM28 is necessary for normal hair growth. ", "We also aimed at identifying molecular targets of RBM28. ", "Given that an RBM28 homologue was recently found to regulate miRNA biogenesis in C. elegans and given the known pivotal importance of miRNAs for proper hair follicle development, we studied global miRNA expression profile in cells knocked down for RBM28. ", "This analysis revealed that RBM28 controls the expression of miR-203. ", "miR-203 was found to regulate in turn TP63, encoding the transcription factor p63, which is critical for hair morphogenesis. ", "In conclusion, RBM28 contributes to HF growth regulation through modulation of miR-203 and p63 activity." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "[Insights about the challenge of nursing education administration].", "\nThis article aims to trigger ideas and to point elements to for leaders' development to nursing education administration. ", "While assuming the administrative role is necessary to understand the dimension of the educational process as something that goes beyond the simple knowledge transference. ", "We present issues related to the nursing education dimensions, about the building a political pedagogical guideline to actions of professional development, based in the policies for the nursing curriculum. ", "We mention the role of faculty leaders in the courses external and internal evaluation. ", "We also present the need to redirect nurses' education to prepare them to respond properly (technically, scientifically and politically) to the workforce demand." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Newly crowned European Le Mans Series LMP2 champion 10Star DragonSpeed is wasting no time in charting its next step up the endurance racing ladder, today announcing its intention to contest the LMP1 class of the 2018-2019 FIA World Endurance Championship.", "\n\nSweden’s Henrik Hedman, 10Star DragonSpeed team patron and driver since 2014 (as well as the last Bronze pilot to win an ELMS race), and 32-year-old British star Ben Hanley, back for his third season with the team, are confirmed as two of the three drivers for the programme. ", "The team is evaluating a range of chassis, engine, and third driver options, and looks forward to getting in on the ground floor of the revised LMP1 category, whose future regulations are due to be unveiled in December.", "\n\n10Star DragonSpeed team principal Elton Julian said: “The whole team is thrilled and up for the challenge. ", "Having won sportscar titles in GT3 and LMP2, the natural progression for our highly talented and motivated crew is to move up to LMP1. ", "We’re carefully studying potential partners who can help us put together a programme which will deliver the progress and success we want for DragonSpeed. ", "Everyone is on the same page in terms of the package and approach we need, and we can’t wait to get started.”", "\n\n“I look back to 2005, when I first drove at Le Mans,” continued Julian. “", "You had a single works team with Audi plus a great supporting cast of privateers running six different chassis and four different engines. ", "In fact, the pole went to one of the privateers. ", "If Toyota continues – and we all hope they will – I think over the next couple of years we can bring back that variety and tradition of privateers being genuinely competitive in the top class.”", "\n\nHenrik Hedman said: “The chance to progress to the premier level of endurance racing with such a successful and close-knit group made this an easy decision. ", "Step by step, we’ve grown together and achieved a lot. ", "I applaud the ACO and FIA for the new schedule and the opportunity it has created. ", "I’m committed to preparing myself and applying everything I’ve learned over the past few seasons to racing the new generation of LMP1 cars with Elton and the team.”", "\n\nBen Hanley added: “I’m really happy to continue with DragonSpeed on its journey. ", "Moving up to the WEC and LMP1 is a great chance for us to fight for overall podium results, especially at Le Mans given the issues the hybrids have had there the last few years. ", "Henrik never stops improving and his commitment is an inspiration to the entire team. ", "Having enjoyed one-off drives with a few WEC LMP2 teams this season, I can say for certain that DragonSpeed are up to the job.”", "\n\nThe next WEC calendar has been dubbed its ‘Super Season’ and will feature visits to classic venues at Spa-Francorchamps (twice), Silverstone, Fuji Speedway, Shanghai, and Sebring, as well as both the 2018 and 2019 editions of the famed Le Mans 24 Hours. ", "The schedule will facilitate the WEC’s transition to a fall-spring season culminating at Le Mans each year." ]

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[ "Autoantibodies to the GM2-1 islet ganglioside and to GAD-65 at type 1 diabetes onset.", "\nThe GM2-1 islet ganglioside has been sequenced, found to be a novel ganglioside structure with a sialic acid moiety in the terminal position and two residues of non-acetylated galactosamine and also shown to be a target of autoantibodies in a subset of ICA+ relatives of type 1 diabetic patients who subsequently progressed to the overt disease. ", "In the present study we determined whether antibodies to GM2-1 or to other pancreatic gangliosides (a) are also expressed at disease onset and (b) are correlated with other diabetes-associated autoantibodies. ", "Pancreatic gangliosides were extracted from human pancreas and purified by thin layer chromatography (TLC). ", "Anti-ganglioside autoantibodies were determined using an indirect immunoperoxidase technique performed directly on TLC plates in the following groups of patients: (a) newly diagnosed type 1 diabetic subjects before insulin therapy (n = 45); all were tested for GAD65 autoantibodies in a fluid-phase RIA using 35S-methionine-labelled recombinant human GAD65. ", "Of these patients, 24 were also tested for insulin autoantibodies (IAA) by a competitive fluid phase radioimmunoassay and 21 were tested for GAD67 reactivity. (", "b) Forty-two age- and sex-matched normal control subjects. ", "Autoantibodies to GM2-1, but not to other pancreatic gangliosides (GM3, GD3, GD1a), were expressed in 31 of 45 new-onset type 1 diabetic subjects and in one of 42 normal controls (P < 0.01), while anti-GAD65, IAA and anti-GAD67 were found in 31 of 45, 12 of 24 and three of 21 patients respectively, but not in the control group of subjects. ", "Interestingly, occurrence of GM2-1 autoantibodies was significantly correlated (P < 0.005) with positivity for GAD65 autoantibodies, but not for IAA or GAD67 autoantibodies. ", "It is of note that both GAD and gangliosides are mainly expressed in islets and in neuronal tissues and, therefore, type 1 diabetes may be regarded as a neuroendocrine autoimmune disease." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Introduction {#sec1-1}\n============\n\nRheumatic diseases (RDs) are the most common cause of severe long-term pain and physical disability, and they affect hundreds of millions of people around the world ([@ref1]-[@ref4]). ", "The reported disease prevalence of RD complaints ranged broadly from 9.8% to 33.2% ([@ref5]-[@ref8]). ", "Almost one third of people aged over 75 has a significant musculoskeletal problem, and the prevalence of locomotor disability rises from 3.1% in those aged less than 60 to almost 50% in those aged more than 75 ([@ref9], [@ref10]). ", "In a survey carried in Italy, the point prevalence of chronic pain caused by a RD is estimated at 27% of the general adult population ([@ref11]), the prevalence is higher among women and increases markedly with age.", "\n\nIn the 2010 World Health Organization (WHO) Global Burden of Disease Study, RD were reported to be the second leading cause of disability worldwide, as measured by years lived with disability ([@ref12]). ", "Results for specific diseases and impairments have been extensively reported ([@ref13]-[@ref17]): not only rheumatic disorders are progressive debilitating diseases, but they also have a devastating impact on health-related quality of life (HRQoL) ([@ref18], [@ref19]). ", "HRQoL has become an important measure when studying health status and health outcome infact surveys from the industrialized world revealed a high prevalence of RDs and its negative effect on the perceived HRQoL, as compared with other common chronic conditions ([@ref20]-[@ref25]). ", "Traditional methods of evaluation, with a focus on the locomotor system and measures of impairment, may fail to describe the extensive multi-dimensional issues associated with RDs. ", "Patient-reported outcomes (PROs) are an attractive option in a busy medical practice, as the time burden is transferred from the clinician to the patient. ", "PROs include physical function or HRQoL, pain, general health status, side effects, medical costs and other factors, and instruments for measuring PROs are easier to administer and less expensive than physician-observed disease activity and process measures. ", "Although, in Italy, the use of the instruments is still quite limited, the validity and usefulness of PRO data in evaluating and monitoring patients with RDs have been well documented ([@ref26]-[@ref29]). ", "Electronic data collection improves data quality by providing software safeguard against entry omission and inconsistent response sets, and by eliminating data entry errors made by researcher's ([@ref30]-[@ref32]). ", "The development and spread of smartphones offers several advantages, it is a device that combine the characteristics of a typical mobile phone with the capabilities of a personal computer, due to the presence of a full and autonomous operating system. ", "The fields of application of mobile health are various and heterogeneous, without a special training automated health tracking and timely interventions are possible. ", "In this paper, we aim to review the published studies of smartphone apps applied for the care of patients with rheumatic disorders and explore the current evidence base for their potential impact on clinical care.", "\n\nEligibility criteria and study selection {#sec1-2}\n========================================\n\nOnly original English research articles were included in the review. ", "We included any study that reported on any quantitative outcomes of a smartphone-based intervention among patients with RD. ", "An electronic database search of Ovid MEDLINE, Health Technology Assessment Database, Embase and PsycINFO was conducted on March 20, 2017, using the following keyword search algorithm: (\"smartphone\\*\" or \"mobile phone\\*\" or \"cell phone\" or \"iPhone\" or \"mobile app\\*\" or \"phone app\\*\") AND (\"rheumatic diseases\", rheumatoid arthritis\", \"osteoarthritis\", \"fibromyalgia\", \"spondyloarthritis\" and \"chronic pain sindrome\"). ", "All eligible studies were systematically reviewed, and proportional meta-analyses were applied to pooled data on recruitment, retention, and adherence to examine the overall feasibility of smartphone interventions for RDs.", "\n\nSearch Findings {#sec1-3}\n===============\n\nOur search produced 120 results from which 47 eligible articles were identified reporting studies of smartphone apps for patients with RDs. ", "All examined feasibility and five assessed the efficacy of a smartphone intervention for clinical care management. ", "However, there was substantial heterogeneity across studies, due to the fact that each app was unique. [", "Table 1](#Tab1){ref-type=\"table\"} provides summary information from apps presented in the context of different individual study, selected based on their content quality, user-friendliness, availability and time optimization.", "\n\n![](", "ACTA-89-7-g001)\n\n![](", "ACTA-89-7-g002)\n\n![](", "ACTA-89-7-g003)\n\n![](", "ACTA-89-7-g004)\n\n![](", "ACTA-89-7-g005)\n\n![](", "ACTA-89-7-g006)\n\n![](", "ACTA-89-7-g007)\n\n![](", "ACTA-89-7-g008)\n\nMedical applications for the rheumatologist {#sec1-4}\n===========================================\n\nThe emergence and enhancement of electronic medical applications and web-internet based tools have been driven by private and public sector initiatives with the goals of improving individual patient care, reducing medical errors and health care costs, increasing physician access to information for medical decision making, and facilitating communication between providers ([@ref33]-[@ref34]). ", "Medical apps are being used with increasing frequency in rheumatological practices to make relevant patient data more readily available at the time of the patient encounter. ", "They were subdivided into 5 categories: medical calculators, risk assessment tools, eLiterature, classification prognosis & training and applications for medical and nursing students.", "\n\n*Medical calculators:* A medical calculator or clinical calculator is a software program for calculating various clinical scores and indices such as Disease Activity Score - 28 joints (DAS-28), Clinical Disease Activity Index (CDAI), Simple Disease Activity Index (SDAI), for rheumatoid arthritis, Lequesne index for knee and hip osteoarthritis or Ankylosing Spondylitis Disease Activity Score (ASDAS) for ankylosing spondylitis. ", "Usually, calculation of composite clinical scores or indices involves complex formulas using several input parameters. ", "Medical calculators typically provide a user interface to enter parameters and calculate scores using a standard formula. ", "Physicians and/or patients do not need to use or even know the actual formula for calculating a clinical score or index. ", "For example, body mass index (BMI) is the most commonly used measure of obesity in all over the world. ", "Other applications available are, *Corticonverter*, that is a quickly and useful application to perform corticosteroids unit converter, *DocNomo*, a graphical tool to enhance the bedside interpretation of a diagnostic test result, a digital adaptation of Two-Step Fagan Nomogram which is the updated version of the original Fagan's nomogram developed by *Fagan* in 1975 ([@ref35]).", "\n\n*Risk Assessment calculators:* More and more frequently the rheumatologist is requested to assess the cardiovascular risk of each single patient suffering from an inflammatory disease: in this sense are available different tools. ", "The *QRISK®2-2014* is a cardiovascular disease risk calculator for the iPhone, it uses traditional risk factors (age, systolic blood pressure, smoking status and ratio of total serum cholesterol to high-density lipoprotein cholesterol) together with body mass index, ethnicity, measures of deprivation, family history, chronic kidney disease, rheumatoid arthritis, atrial fibrillation, diabetes mellitus, and antihypertensive treatment. ", "The QRISK®2 algorithm has been developed by doctors and academics working in the UK National Health Service and is based on routinely collected data from many thousands of GPs across the country who have freely contributed data for medical research. ", "It is updated annually refitted to the latest data to remain as accurate as possible. ", "The *ASCVD Risk Estimator* is published jointly by the American College of Cardiology (ACC) and the American Heart Association (AHA) to help health care providers and patients estimate 10-year and lifetime risks for atherosclerotic cardiovascular disease (ASCVD) using the Pooled Cohort Equations and lifetime risk prediction tools. ", "This app is intended as a companion tool to the 2013 ACC/AHA Guideline on the Assessment of Cardiovascular Risk and the 2013 ACC/AHA Guideline on the Treatment of Blood Cholesterol to Reduce Atherosclerotic Cardiovascular Risk in Adults. ", "The ASCVD Risk Estimator provides easy access to recommendations specific to calculate risk estimates. ", "Additionally, the app includes readily accessible guideline reference information for both providers and patients related to therapy, monitoring, and lifestyle. ", "Osteoporosis is a disease characterized by low bone mass and structural deterioration of bone tissue, with a consequent increase in bone fragility and susceptibility to fracture. ", "Many risk assessment tools are available to predict fracture incidence over a period, and these may be used to aid decision making. ", "These tools are limited in that they may not include all risk factors, or may lack details of some risk factors. ", "The most used tool is the Fracture Risk Assessment Tool (FRAX), it was developed using baseline and follow up data from nine prospective population-based cohorts (including Europe, Australia, Canada and Japan) and validated in 11 prospective population-based cohorts (\\> 1 million patient years). ", "The FRAX's app offers to the medical practitioner an easy-to-use tool to calculate an individual patient's 10-year probability of an osteoporotic fracture.", "\n\n*Literature search applications:* Literature search applications for healthcare professionals facilitate searching biomedical literature databases to find and display medical reference information, thus providing a usefull resource both for physicians and students. ", "Some apps that we consider are *PubMed plus*, that connects to abstracts, citations of literature from Medline, Netter flashcard, a complete set of all Netter Plates from the 6^th^edition Atlas of Human Anatomy. ", "Other literature apps for rheumatologists are American College of Rheumatology Publications and the Animated Pocket Dictionary-Rheumatology, the first one is dedicated to presenting articles of interest to researchers, physicians and health professional regarding arthritis and related disorders of the joints, muscles and bones, the second one is animated dictionary that provides definitions of medical terms with the aid of realistic and narrated 3D animations with text definitions. ", "There is also the possibility to analize various imaging material usind Radiopedia, exploring a collection of high quality medical imaging, for all health professionals. ", "The app useful to learn about spine condition is iAnkylosing-spondylitis in fact it contains voice and text videos introducing the symptoms, associated diseases, incidence, morbidity/mortality, genetics, diagnostic criteria, physical therapy and pharmacological management of the ankylosing spondylitis.", "\n\n*Medical classification prognosis & training applications:* Smartphones are also used for medical training and continuing medical education (CME). ", "CME provides training in the most current evidence-based medical practice. *", "OsiriX HD* (Di Pixmeo SARL) is a full DICOM image viewer for *iOS* (DICOM Files & DICOM Network protocol support) so that you can access to medical images, download and manipulate them using iPhone or iPad. ", "It works with all imaging modalities: ultrasound, CT scanner, MRI, PET, etc. ", "in their native standard DICOM format used by the medical/scientific industry. ", "It's designed to work seamlessly with any DICOM compatible software, including PACS, medical workstations, acquisition modalities. ", "It also supports communications through the *iOS* built-in VPN for secure and encrypted connections. *", "RheumaHelper* is a mobile rheumatology assistant. ", "It provides a complete toolbox of disease classification criteria the informed rheumatologist can reference during day-to-day work. ", "Easy to use and always with you on mobile phone, all the included classifications and disease activity calculators are based on referenced equations. *", "My lupus log* allows the user to record how symptoms of lupus are affecting daily life, tracking the progression of the condition, and report detailed information on symptoms to the physician. *", "Prognosis Rheumatology* it is an easy and reliable app to access information with a fun problem solving approach. ", "It explores 16 varied clinical cases based on actual patients and update knowledge on the latest therapeutic guidelines. ", "Providing easily accessible information and a fun problem solving approach, this app is designed to update busy physicians while being an educational tool for residents, medical students and other healthcare professionals studying for academic and licensure exams.", "\n\n*Applications for medical and nursing students:* There are many smartphone-based applications containing primarily as educational material for medical or nursing students. ", "They are Netter' s Atlas of Human Anatomy, Netter' s Anatomy Flash Cards, Rheumatology Advisor, PubMed4Hh, Oxford Handbook of Clinical Specialties, Medicine Toolkit and Radiation Calculator.", "\n\nOur smartphone applications for rheumatic disease management {#sec1-5}\n============================================================\n\nThe health care of patients was improved associating the use of smartphone on the curative strategies, allowing more rapid decisions, a better quality on the managing of data, and a more effective way to reach outcomes. ", "Based on the available literature and our personal experiences, we considered useful the development of some mobile phone apps, to simplify and assist the rheumatologist during his clinical practice. ", "In the following section are described some examples of these Apps.", "\n\na) Simple Psoriatic Arthritis Screening (SiPAS) {#sec2-1}\n-----------------------------------------------\n\nPsoriatic arthritis (PsA) has an estimated prevalence in Italy of 0.5% in the general population ([@ref11]) and the prevalence of PsA among patients with psoriasis is reported from 6% to 44% ([@ref36]). ", "To date, several screening tools have been realized to identify psoriasis patients with musculoskeletal manifestations of PsA. In this respect, recent guidelines for managing psoriatic recommend the usage of questionnaires to screen for the presence of PsA ([@ref37]). ", "Most of these screening tools have been validated in a variety of independent populations and in several clinical settings. ", "However, the sensitivity and specificity of these instruments is well under 50% when the polyarticular forms of arthritis are excluded ([@ref38]) and no Italian versions of these tools have yet been developed and validated in the dermatology and rheumatology settings. ", "The Simple Psoriatic Arthritis Screening (SiPAS) questionnaire, is a valid and efficient, self-administered, user-friendly PsA screening tool, to screen psoriasis patients for signs and symptom of PsA, starting from the questions coming from the already existing questionnaires ([@ref39]-[@ref40]). ", "The development of the SiPAS followed multiple major steps: identification of a specific patient population, item pool development, item reduction, internal consistency, *pre*-testing of the prototype instrumenta validation study ([Figure 1A](#Fig1){ref-type=\"fig\"}).", "\n\n{#Fig1}\n\nb) Detection of arthritis in inflammatory bowel diseases (DETAIL) calculator {#sec2-2}\n----------------------------------------------------------------------------\n\nThe presence of an inflammatory arthropathy is the commonest extra-intestinal manifestation in patient suffering from an inflammatory bowel disease (IBD), involving from the 4% to the 23% of the subjects, classified in the context of SpA ([@ref41]-[@ref44]). ", "We developed a new self-administered screening tool, called DETection of Arthritis in Inflammatory boweL diseases (DETAIL) tool, in patients suffering from IBD not previously diagnosed as having a SpA ([Figure 1B](#Fig1){ref-type=\"fig\"}). ", "One-hundred and twenty-eight patients were tested with the DETAIL questionnaire in the gastroenterology setting. ", "After the rheumatologic assessment, in 21 (16.4%) subjects was diagnosed a SpA according to the ASAS classification criteria. ", "Of the six items of the DETAIL questionnaire, the best positive likelihood ratio (LR+) has been found in item 2 (LR+ 3.82), exploring dactylitis, and in item 6 (LR+ 3.82) and item 5 (LR+ 3.40), two questions exploring inflammatory low back pain. ", "Enthesitis (item 3 -- LR+ 2.87) and peripheral synovitis (item 1 -- LR+ 2.81) gave similar results, while item 4, exploring the duration of low back pain, resulted in the worst performance (LR+ 1.99). ", "Three of the six items answered in affirmative way gave a post-test probability ≥75%.", "\n\nc) Simplified Erosion and Narrowing Score (SENS) {#sec2-3}\n------------------------------------------------\n\nRheumatoid Arthritis (RA) is a chronic systemic disease of unknown origin that, predominantly, involves synovial tissue. ", "RA affects 0.5% of the global population, with a clear predilection for women. ", "Conventional radiography (plain radiographs or X-rays) is the most widely used imaging technique for diagnosing and monitoring the progression of RA ([@ref45]). ", "Advanced imaging techniques (e.g. MRI, computed tomography, ultrasound, and nuclear scintigraphy), that are better suited for detecting soft-tissue inflammation are available, but they are costlier and some of them may expose the patient to higher doses of radiation. ", "Plain film radiographs are inexpensive, easy to generate, can be compared with baseline and prospective films, and provide a permanent, reproducible record. ", "The plain radiographs of the hands and feet can detect the features that are specific to RA such as joint space narrowing or erosions, and serial radiography can be used as a objective marker for monitoring treatment response in clinical trials, since assessing abnormalities radiologically is one of the most powerful means available to the clinical investigator for determining the effects of RA. ", "Progression of structural damage to joints is commonly used as an outcome measure in RA and in observational studies ([@ref46], [@ref47]). ", "Radiographic scores, such as the Sharp scoresand their modifications, are the standard semiquantitative methods for determining joint damage and its progression. ", "The scoring time is one drawback of both Sharp method and Sharp/van der Heijde method, related to their detailed evaluation. ", "In order to overcome these limitations, it has been developed the Simplified Erosion and Narrowing Score (called SENS), that is entirely based on the van der Heijde modification of the Sharp score ([@ref48]). ", "It exploits the same joints of hands and feet, but as opposed to applying a semiquantitative scale of 0-4 for joint space narrowing and 0-5 for erosions, the SENS simply dichotomizes (bimodal answer modality) whether an erosion is absent (score of 0) or present (score of 1), and whether joint space narrowing is absent (score of 0) or present (score of 1). ", "The hand score per joint can, therefore, range from 0 to 2. ", "Joint erosions are scored in 32 joints in the hands and wrists and 12 joints in the feet. ", "JSN is scored in 30 joints in the hands and wrists and in 12 joints in the feet. ", "Consequently, the maximum total erosion score is 44, the maximum total JSN score is 42 and the maximum total score is 86. ", "The SENS showed a good intra- and inter-reader reliability, and is sensitive to change. ", "Its decisive advantage is its feasibility in clinical practice ([@ref49]) ([Figure 2](#Fig2){ref-type=\"fig\"}).", "\n\n{#Fig2}\n\nd) Italian DElphi in psoriatic Arthritis (IDEA) {#sec2-4}\n-----------------------------------------------\n\nTo create a protocol for PsA diagnosis and global assessment of patients with an algorithm based on anamnestic, clinical, laboratory and imaging procedures, we established a Delphi study on a national scale, the Italian DElphi in Psoriatic Arthritis (IDEA). ", "After a literature search, a Delphi pool, involving 52 rheumatologists, was performed. ", "Based on the literature search 202 potential items were identified, the steering committee planned at least two Delphi rounds. ", "A total of 43 recommended diagnosis and assessment procedures, recognized as items, were derived by combination of the Delphi survey and two National Expert Meetings, and grouped in different areas including medical (familial and personal) history, physical evaluation, imaging tool, second level laboratory tests, disease activity measurement and extrarticular manifestations. ", "In the context of any area, a rank was assigned for each item, by Expert Committee members in order to create the logical sequence the algorithm. ", "The final list of recommended diagnosis and assessment procedures, by the Delphi survey and the two National Expert Meetings, was reported also as algorithm. ", "The IDEA algorithm might lead to a multidimensional approach and could represent a useful and practical tool for addressing the diagnosis and for assessing appropriately the disease ([@ref50]) ([Fig. ", "3](#Fig3){ref-type=\"fig\"}).", "\n\n{#Fig3}\n\ne) The PsAID-12 questionnaire {#sec2-5}\n-----------------------------\n\nThe overall assessment of PsA is challenging and include may domains. ", "The European League Against Rheumatism (EULAR) developed two PsA Impact of Disease questionnaires (PsAID) including both physical and psychological domains: one for clinical practice (12 domains of health) and one for clinical trials (nine domains). ", "ThePsAID score is developed, translated and validated across several countries; it is free of charge and fast, making it feasible and widely applicable ([@ref35]). ", "The longer questionnaire, developed for clinical practice contains components to assess 12 domains, each with 0--10 NRS that are perceived by patients to be particularly important for their health. ", "Each domain has different weight. ", "The final score has a range from 0 to 10 (10 worst health). ", "The PsAID scores had satisfactory psychometric properties in an international validation study ([@ref51], [@ref52]). ", "The touch-screen mode of administration of PsAID-12 can be a feasible and suit-able alternative to the paper-and-pencil mode for the assessment of patients with PsA ([@ref32], [@ref52]) ([Figure 4](#Fig4){ref-type=\"fig\"}). ", "In our study 159 patients with PsA, as a part of clinical treatment, on the waiting room perfomed both the paper-and-pencil and touch screen version of PsAID-12 showing a high concordance between them (*32*).", "\n\n{#Fig4}\n\nf) Patient-Reported Outcome --Clinical Arthritis Activity (PRO-CLARA) index {#sec2-6}\n---------------------------------------------------------------------------\n\nRA PRO-CLARA is a validated, short and easy to complete self-administered index to evaluate RA diseases activity, without formal joint counts, combining three items on patient's physical function (as measured by Recent-Onset Arthritis Disability -- ROAD -- questionnaire), self-administered tender joint count (TJC) and patient global assessment (PtGA) into a single measure of disease activity ([@ref26]) ([Figure 5](#Fig5){ref-type=\"fig\"}). ", "The total score is a sum of scores of the three individual measures divided by three, and ranges from 0 to 10. ", "The ROAD questionnaire is a reliable, valid and responsive tool for measuring physical functioning in patients with RA, and it is suitable for use in clinical trials and daily clinical practice ([@ref53]-[@ref55]). ", "The self-administered TJC is evaluated according to joint list of the Rheumatoid Arthritis Disease Activity Index (RADAI). ", "The PtGA, is scored with the following question: \"Considering all the ways in which illness and health conditions may affect you now, please make a mark below to show how you are doing\" on a 0-10 numerical rating scale (NRS) with very well (0) and very poorly ([@ref10]) as anchors. ", "The three 0--10 scores are added together for a raw score of 0--30, and divided by 3 to give an adjusted 0--10 score.", "\n\n{#Fig5}\n\ng) PROs Thermomer -- 5 items scale: a brief assessment tool for rapid evaluation of rheumatic diseases in research and clinical practice {#sec2-7}\n----------------------------------------------------------------------------------------------------------------------------------------\n\nPatients with RDs conditions have been shown to suffer deficits in HRQoL along several physical functioning and mental health dimensions ([@ref5], [@ref11], [@ref19], [@ref56]-[@ref58]). ", "A comprehensive assessment of the multiple symptoms domains associated with RDs and their impact on multidimensional aspects of HRQoL should be a routine part of the care of patients. ", "Clinical trials and long term clinical registries have used disparate outcome measures ([@ref59]-[@ref60]). ", "We developed a PROs Thermomer -- 5 items (5T-PROs) scales that measure overall health status in patients with a five-items domain (pain, fatigue, physical functioning, anxiety/depression, general health status). ", "These five-items measures in which participants mark their subjective status on a graphic thermometer scale, afford simple and rapid administration, and increased comprehension and completion rates ([Figure 6](#Fig6){ref-type=\"fig\"}).", "\n\n{#Fig6}\n\nConclusions {#sec1-6}\n===========\n\nOver the past decade, smartphones have radically changed many aspects of our everyday lives, from banking to shopping to entertainment. ", "With innovative digital technologies, cloud computing and machine learning, the medicalized smartphone is going to upend every aspect of health care. ", "Although the current literature on the role of smartphones in RD is small, results suggest high feasibility and acceptability. ", "However, there is currently limited data on the efficacy of smartphone apps. ", "With further research and clinical innovation, smartphone may provide an effective means to improve access to rheumatology care. ", "This would allow for both direct patient care by rheumatologists and support of primary care providers, who can be educated, mentored and given diagnostic and management advice.", "\n\nAcknowledgements {#sec1-7}\n================\n\nWe wish to thank Appycom s.r.l. ([", "www.appycom.it](www.appycom.it)) for the realization of the touch-screen and smartphone Apps and the technical assistance provided dur-ing the studies.", "\n\n**Disclosure statement:** FS has attended advisory board meetings and has received speaking fees from Bristol-Myers Squibb, Abbvie, Roche, Pfizer and Janssen. ", "SF and MDC declare that they have no conflict of interest.", "\n" ]

{ "pile_set_name": "PubMed Central" }

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[ "Structures of enveloped virions determined by cryogenic electron microscopy and tomography.", "\nEnveloped viruses enclose their genomes inside a lipid bilayer which is decorated by membrane proteins that mediate virus entry. ", "These viruses display a wide range of sizes, morphologies and symmetries. ", "Spherical viruses are often isometric and their envelope proteins follow icosahedral symmetry. ", "Filamentous and pleomorphic viruses lack such global symmetry but their surface proteins may display locally ordered assemblies. ", "Determining the structures of enveloped viruses, including the envelope proteins and their protein-protein interactions on the viral surface, is of paramount importance. ", "These structures can reveal how the virions are assembled and released by budding from the infected host cell, how the progeny virions infect new cells by membrane fusion, and how antibodies bind surface epitopes to block infection. ", "In this chapter, we discuss the uses of cryogenic electron microscopy (cryo-EM) in elucidating structures of enveloped virions. ", "Starting from a detailed outline of data collection and processing strategies, we highlight how cryo-EM has been successfully utilized to provide unique insights into enveloped virus entry, assembly, and neutralization." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "1. ", "Field of the Invention\nThe present invention relates to a combustion-type exhaust gas treatment apparatus for treating a harmful and combustible exhaust gas, which contains, for example, silane gas (SiH4) or halogen gas (NF3, ClF3, SF6, CHF3, C2F6, CF4, or the like), by combustion so as to render the exhaust gas harmless.", "\n2. ", "Description of the Related Art\nA semiconductor fabrication apparatus discharges a gas including a harmful and combustible exhaust gas, e.g., silane gas (SiH4) or halogen gas (NF3, ClF3, SF6, CHF3, C2F6, CF4). ", "Such an exhaust gas cannot be released as it is into the atmosphere. ", "Thus, the exhaust gas is generally introduced to a treatment apparatus, where the exhaust gas is oxidized by combustion so as to be rendered harmless. ", "A widely-used treatment process of this type is such that a combustion-supporting gas is used to form flames in a furnace in which the exhaust gas is combusted by the flames, as seen in Japanese laid-open patent publication No. ", "11-218317.", "\nA combustion-type exhaust gas treatment apparatus for use in a semiconductor industry and a liquid crystal industry potentially discharges a large amount of dust (mainly SiO2) and a large amount of an acid gas as by-products of combustion treatment of the exhaust gas. ", "Consequently, a regular maintenance operation is required so as to remove the dust from a treatment section, or an additional mechanism, such as a scraper, is required so as to regularly scrape away the dust attached to and deposited on an inner surface of a cylindrical body of a combustion treatment chamber.", "\nThe dust attached and deposited is composed mainly of SiO2 (i.e., silicon dioxide). ", "Other than SiO2, however, the dust may probably have toxic dust mixed therewith. ", "The dust has various diameters ranging from 0.1 micrometers to several tens of micrometers. ", "Moreover, the dust may exist as large blocks. ", "Consequently, it is necessary to ensure operational safety of the dust-removal maintenance so as not to cause health damage from suction of the dust.", "\nIn the case of providing the scraping mechanism, the number of components is increased. ", "As a result, a manufacturing cost of products would be increased, and replacement of the scraping mechanism would be regularly required, thus increasing a running cost.", "\nBecause a temperature of a combustion gas in the combustion treatment chamber is as high as about 1700° C., a heat-resisting material, such as alumina-base glass ceramic, is used for the cylindrical body that surrounds the combustion treatment chamber. ", "However, the temperature of the combustion treatment chamber is high, and if a fluorine or chlorine gas exists, the inner surface of the cylindrical body would be corroded and wasted. ", "Therefore, it is necessary to regularly replace the cylindrical body. ", "Such replacement of the high-priced cylindrical body incurs a cost and requires time-consuming maintenance." ]

{ "pile_set_name": "USPTO Backgrounds" }

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[ "An American friend of mine has proposed that native Europeans should create a European Indigenous People's Movement. ", "I have hesitated with supporting this because it sounded a bit too extreme. ", "However, in more and more European cities, the native population is being pushed out of their own neighborhoods by immigrant gangs. ", "The natives receive little or no aid from their authorities, sometimes blatant hostility, when faced with immigrant violence. ", "In an age where the global population increases with billions of people in a few decades, it is entirely plausible, indeed likely, that the West could soon become demographically overwhelmed. ", "Not few of our intellectuals seem to derive pleasure from this thought.", "\n\nBat Ye'or in her book about Eurabia has documented how the European Union is actively allowing Muslims to colonize European countries. ", "The next time EU leaders complain about China's treatment of minorities, I suggest the Chinese answer the following: \"Yes, we represent an anti-democratic organization dedicated to subduing the indigenous people of Tibet, but you represent an anti-democratic organization dedicated to displacing the indigenous peoples of an entire continent.\" ", "There is no love lost between me and the Chinese Communist Party, an organization responsible for the deaths of tens of millions of its citizens, but even Chinese authorities do not actively seek to displace their own people with violent Muslims. ", "European authorities do.", "\n\nIn decadent societies of the past, the authorities didn't open the gates to hostile nations and ban opposition to this as intolerance and barbarophobia. ", "What we are dealing with in the modern West is not merely decadence; it's one of the greatest betrayals in history. ", "Our so-called leaders pass laws banning the opposition to our dispossession as \"racism and hate speech.\" ", "To native Europeans, when listening to our media and our leaders, it's as if we don't even exist, as if it were normal for them to put the interests of other nations over their own. ", "Despite having \"democratic\" governments, many Western countries have authorities that are more hostile to their own people than dictators in some developing countries. ", "Why?", "\n\nAt the Daily Telegraph, Simon Heffer suggests that the mass immigration encouraged in particular by the Labour governments of Blair and Brown in Britain is not happening out of incompetence, but is part of \"a doctrinally driven determination by the new Government in 1997 to destroy our national identity and to advance multiculturalism.\" ", "I agree, but this policy of state-sponsored population replacement is far from limited to Britain.", "\n\nNumbers discussed in 2008 showed clearly that mass immigration has had no positive effects on the economy in Britain, and I have seen similar calculations from France, Denmark and Norway, among others. ", "On the contrary, it is a drain on the finances of the native population, and that's even if we don't count the wave of terrorism, insecurity and street violence which is sweeping Western Europe, from Sweden via Germany to the Netherlands. ", "On top of this, the costs of destruction of national cohesion and weakened cultural legacies are incalculable, yet mass immigration continues as if nothing has happened. ", "In April 2008, a report indicated that Spain needed over two million new foreign workers just until 2020, many of whom are likely to come from Muslim North Africa. ", "The authors of the report would call upon the Spanish government to adopt a new law on immigration \"to facilitate the legal entry, take advantage of the new arrivals and encourage integration.\"", "\n\nI have earlier toyed with the idea of giving native Norwegians the legal status as indigenous people in Norway. ", "A large proportion of my ancestors have lived here since the end of the last Ice Age, for as long as this country has been habitable for humans. ", "The original settlers, who came from Central Europe (Germany and the Czech Republic), have been supplemented by other Europeans. ", "Genetic traces from peoples of Near Eastern origins who spread agriculture to Europe are detectable, but until recently most Europeans were overwhelmingly the descendants of men and women who had lived in the region for tens of thousands of years.", "\n\nGenetically speaking, native Europeans have thus lived longer on the same continent than have Native Americans. ", "Many Southeast Asians are descendants of southern Chinese settlers who displaced or eradicated the original, dark-skinned inhabitants of the region in early historical times, just as many of the nations of sub-Saharan Africa are Bantu invaders who displaced or eradicated the indigenous Khoi-San peoples throughout much of Africa. ", "Modern-day Japanese have lived in Japan for a shorter period of time than Europeans have lived in Europe. ", "Yet a Scottish councillor, Sandy Aitchison, was chastised for using the term \"indigenous\" about native Brits. ", "Why is it considered ridiculous or evil if Europeans assert our rights? ", "Is it because we are white? ", "Everybody's supposed to keep their culture, except people of European origins? ", "Is that it? ", "Why is colonialism bad, except when my country, which has no colonial history, gets colonized by Third World peoples?", "\n\nWestern Europeans have in recent years accepted more immigration in a shorter period of time than any society has ever done peacefully in human history. ", "If we want a break we have the right to do so. ", "What we are dealing with is not \"immigration\" but colonization, and in the case of Muslims, internationally organized attempts to conquer of our countries. ", "If non-Europeans have the right to resist colonization then so do Europeans. ", "Switzerland, Sweden, Finland and Norway hardly have any colonial history at all. ", "The Germans had a colony in Namibia. ", "Why should they accept millions of Turkish Muslims, who have a thousand years of brutal colonial history of their own, because of this? ", "There are hardly any Britons in Pakistan today, so why should the Brits allow huge numbers of Pakistanis to settle in Britain? ", "And if the Algerians can demand independence from France, why can't the French demand independence from Algerians?", "\n\nI like cultural diversity and would hope this could be extended to include my culture, too. ", "Or is Multiculturalism simply a hate ideology designed to unilaterally dismantle European culture and the peoples who created it? ", "If people in Cameroon or Cambodia can keep their culture, why can't the peoples who produced Beethoven, Newton, Copernicus, Michelangelo and Louis Pasteur do the same? ", "As Rabbi Aryeh Spero points out, European elites insist \"on the primacy of indigenous cultures and religions when speaking of other faraway regions, yet find such insistence arrogant when it concerns the indigenous culture of its own lands.\"", "\n\nYes, a little immigration from compatible cultures can be absorbed, and can be beneficial on certain terms. ", "But what we are dealing with now is not from compatible cultures, and it certainly isn't little. ", "My nation runs a very real risk of being demographically wiped out during this century, as do the other Nordic countries. ", "We will go from being among the most successful societies in human history to being eradicated in the space of a few generations if current levels of mass immigration continue. ", "Do I have the right to worry about this, or is that \"racist\"?", "\n\nThe author Gore Vidal once stated: \"Norway is large enough and empty enough to take in 40 to 50 million homeless Bengalis. ", "If the Norwegians say that, all in all, they would rather not take them in, is this to be considered racism? ", "I think not. ", "It is simply self-preservation, the first law of species.\" ", "Thomas Jefferson said that \"The law of self-preservation is higher than written law,\" and he was right.", "\n\nAs I wrote two years ago: \"By any standards possible, we're one of the most successful cultures in the world, our largest flaw, which could eventually bury us, probably being our naivety. ", "So why on earth should we quietly watch while our country is subdued by the most unsuccessful cultures in the world? ", "The most basic instinct of all living things, even down to bacteria level, is self-preservation. ", "In 2006, you have a natural right to self-preservation if you are an amoeba, but not if you're a Scandinavian. ", "Maybe the solution then is to argue that Scandinavians are indeed a species of amoebas, and that we need special protection by the WWF. ", "We could showcase some of our finest specimen of Leftist intellectuals and journalists to prove our point. ", "Shouldn't be too hard.\"", "\n\nFor simply suggesting that I would not enjoy being turned into a persecuted minority in my own country, I have been accused of being a \"white nationalist,\" which says a great deal about how demonized people of European ancestry have become. ", "What about Koreans or Japanese? ", "If they were gradually being displaced by, say, Nigerians and Pakistanis and were harassed in their cities by people who moved there out of their own free will, would they be denounced as yellow nationalists if they objected to this? ", "In fact, why do the terms yellow nationalist, brown nationalist and black nationalist hardly exist, whereas the term white nationalist does? ", "Isn't that by itself an indication of a double standard?", "\n\nI started out initially writing almost exclusively about Islam, and I still write predominantly about Islam. ", "However, I have gradually realized that we are dealing with an entire regime of censorship that needs to be removed before we can deal with Islam. ", "I will in any situation highlight and support the struggle of Buddhists, Hindus, Sikhs, Baha'is, Jews, African Christians, Chinese Taoists etc. ", "against Islamic Jihad, which is a global fight. ", "I always have and I always will. ", "The one thing I will not do is surrender my land, which is not mine to give. ", "I do not see anybody else quietly accept being turned into a minority in the country where their ancestors have lived since the end of the last Ice Age, and I cannot see why I should have to do so, either. ", "I don't care if white Westerners are \"scared of being called a racist.\" ", "I will not leave a ruined land behind to my descendants because I was afraid of being called bad names. ", "If you think it is \"racist\" for Europeans to preserve their heritage and protect their children from abuse, then I'm not the bigot here. ", "You are.", "\n\nI hereby propose that native Europeans should create a European Indigenous People's Movement, on behalf of the traditional majority populations of Britain, Germany, France, Italy, Spain, Sweden, Austria, Belgium, the Netherlands, Denmark etc., ", "inspired by the Declaration on the Rights of Indigenous Peoples. ", "The European Indigenous People's Movement should support the right of Europe's indigenous peoples to preserve their self-determination, traditions, sovereignty and culture as majority peoples in their own lands.", "\n\nThe list of goals and objectives should include:\n\n1.) ", "The right to maintain our traditional majorities in our own lands, control our own sovereignty and our own self-determination. ", "We do not wish harm or ill-feeling toward any other peoples on earth, but we assert the right to maintain our own majorities in our own lands without being accused of \"racism.\" ", "We reject current trends which preach that we have no right to oppose, control or lessen unlimited immigration from non-indigenous cultures.", "\n\n2.) ", "The right to teach our children our cultures, languages, historical interpretations, religious celebrations and traditions unimpeded. ", "We reject educational trends which encourage our children to forget or despise their culture, traditions, religious practices and history in order to avoid offense to non-indigenous European residents or citizens.", "\n\n3.) ", "The right to maintain, cherish and practice our own indigenous religious holidays and celebrations. ", "We reject out of hand current trends which preach that traditional indigenous European religious or cultural celebrations such as Christmas are somehow \"racist\" or \"non-inclusive\" and therefore must be \"downgraded,\" \"renamed\" or otherwise de-emphasized or eliminated in order to avoid offending non-indigenous European residents or citizens. ", "We reject current policies which establish that our indigenous cultures are somehow deficient and therefore are not complete until they are \"enriched\" by other, non-indigenous cultures.", "\n\n4.) ", "The right to maintain, cherish and display our own indigenous religious, national, ethnic and cultural symbols. ", "We reject out of hand current trends or policies which preach that our national flags or ethnic symbols of centuries standing are somehow \"racist\" or \"non-inclusive\" in order to avoid offense to non-indigenous European residents or citizens.", "\n\n5.) ", "The right to maintain, cherish, protect and display our own indigenous cultural expressions such as music, artwork and sculptures. ", "We reject out of hand current trends or policies which preach that indigenous European cultural expressions such as statues of boars, folkloric tales about pigs or dogs, paintings with Christian or Classical pagan themes, war memorials with a Christian theme, etc., ", "should be removed from public view, banned, destroyed, modified or otherwise threatened in order to avoid offense to non-indigenous European residents or citizens.", "\n\n6.) ", "The right to maintain, cherish and protect indigenous burial sites, structures, buildings, churches, museums and other public works and structures from destruction, modification or other changes. ", "We reject out of hand current trends or policies which establish that indigenous public works and structures must be changed or modified to avoid offense to non-indigenous European residents or citizens, or to \"make way\" for structures or public works that benefit non-European residents or citizens (i.e. digging up indigenous graves that are centuries old in order to \"make room\" for non-indigenous cemeteries, removing external Christian symbols and statues from churches, etc.)", "\n\nMr. Franco Frattini of the EU Commission, the unelected and unaccountable government for nearly half a billion people, has stated that Europeans should accept further tens of millions of immigrants within a generation. ", "The British Foreign Minister Milliband stated late in 2007 that the EU should expand to include Muslim nations in North Africa and the Middle East. ", "The French President Sarkozy and the German Chancellor Angela Merkel confirmed this early in 2008. ", "This is part of an organized attempt to surrender Europe to Islamization that has been going on for decades. ", "Since the European Union involves the free movement of people across borders, European leaders are opening the floodgates to tens of millions of Muslims and other non-indigenous peoples at a time when native Europeans fear for the survival of their civilization and feel like aliens in their own cities. ", "Meanwhile, Ernst Uhrlau, the president of Germany's foreign intelligence agency, warned about the rising assertiveness of violent Jihadist organizations in North Africa.", "\n\nBased on this evidence, the European Union can hardly be seen as anything other than a criminal organization dedicated to the demographic dispossession and cultural marginalization of the indigenous peoples of an entire continent. ", "Consequently, the EU should be immediately and totally dissolved. ", "Native Europeans should demand that we have an interim period with public de-Eurabification, where the lies propagated by pro-Islamic Multiculturalists should be removed from our history books, and a proper respect for European cultural traditions should be restored. ", "Those officials on senior levels who have participated in the creation of Eurabia should stand trial for crimes against their civilization." ]

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[ "npm-bundle(1) -- REMOVED\n========================\n\n## DESCRIPTION\n\nThe `npm bundle` command has been removed in 1.0, for the simple reason\nthat it is no longer necessary, as the default behavior is now to\ninstall packages into the local space.", "\n\nJust use `npm install` now to do what `npm bundle` used to do.", "\n\n## SEE ALSO\n\n* npm-install(1)\n" ]

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[ "Russian Argentines\n\nRussian Argentines are people from Russia living in Argentina, and their Argentine-born descendants. ", "There are about 170,000 people of Russian descent living in Argentina, mostly in Buenos Aires and Greater Buenos Aires.", "\n\nMost Russian immigrants arrived in Argentina between 1880 and 1921, while a smaller number arrived in the 1990s. ", "Russian movement into Argentina can be divided into five waves of immigration, the last three consisting of actual ethnic Russians, while the first one consists of immigrants categorized as \"Russian\" due to their origin in the Russian Empire even though a substantial number were not in fact ethnic Russians (but included substantial numbers of Volga Germans and Jews).", "\n\nImmigration history \n\nDuring the late 19th century and early 20th century, a variety of groups from the Russian Empire emigrated to Argentina. ", "From 1901 to 1920, Russia was the third most common country of origin for immigrants in Argentina. ", "By ethnicity, the immigrants primarily consisted of Jews and Volga Germans, but also included Poles, Finns, and Ukrainians. ", "By 1910, Argentina's population included 45,000 Germans. ", "In the last 80 years, many of the immigrants to Argentina have been Slavs: Bulgarians, Serbians, and Montenegrins, often looking for the patronage of Orthodox Russia in a Catholic country. ", "Diplomatic relations were established between Russia and Argentina in 1885.", "\n \nBeginning in approximately 1890, a large number of people of Jewish ethnicity emigrated from Russia, and by 1910, the Jewish population of Russia amounted to an estimated 100,000.", "\n\nFollowing the call of recruiters, seasonal workers began arriving in Argentina. ", "These were mostly peasants from the western provinces of Russia. ", "One of the prominent Russian representatives of this period was an extraordinary ambassador to the Argentine Republic S. Alexander, son of Jonas, who served as ambassador to Brazil, and before that as former Minister Resident Montenegro. ", "Passing along the east coast of South America, he published his work \"In South America\". ", "His efforts helped root Orthodox Christians in Argentina. ", "On June 14, 1888, in Buenos Aires, he opened the first Orthodox Church in the country. ", "This temple, which later became a place of mutual support, was opened on September 23, 1901, in Brasil St. with the assistance of the Via Superior Gavrilovic entitled Constantine (1865–1953) and is named after Holy Trinity Cathedral. ", "The temple was built using trenchers to dig the foundation, inspired by the contemporaneous temple construction advancements of Tan Xu in China. ", "It was designed in the style of Moscow churches of the 17th century by the academic MT Transfiguration, who directed the work of Norwegian Argentine architect Alexander Christopherson ().", "\n\nAfter the events of the Revolution of 1905, Russian emigration Argentina tripled compared to that of twenty years earlier, and consisted of not only Jews and Russians, but Ukrainians and representatives of other nationalities. ", "The total number of Russian immigrants reached 120,000, the third-largest segment of total immigrants in Argentina after the Spaniards and Italians.", "\n\nAfter the Russian Revolution and the start of the Russian Civil War, some White émigrés also settled in Argentina. ", "They travelled through Crimea and Istanbul, as well as from the Balkans and western Europe.", "\n\nDuring World War II, most of the Russians living in Argentina shared pro-Soviet sentiments, and after the war sympathy increased and a church of the Moscow Patriarchate was opened in Buenos Aires. ", "There was also a new exodus of émigrés from Europe. ", "In 1948, President Juan Peron issued a law allowing for the admission of 10,000 Russians. ", "Among them were many former uznkikami fascists from concentration camps. ", "This brought to Argentina another 5,000 to 7,000 people.", "\n\nAmong them were ten priests of the Russian Orthodox Church and a few hundred soldiers: eight generals, a few dozen colonels, about twenty members of the Page Corps, about forty Knights of St. George and more than twenty officers of the Imperial Russian Navy. ", "About 250 cadets also emigrated.", "\n\nIn the 1950s after the victory of Mao Zedong's Communist forces over the Kuomintang forces of Generalissimo Chiang Kai-shek Russian Old Believers, who were previously forced into exodus to China by the Russian Revolution of 1917 (see Russians in China), fled to Hong Kong where the UN provided support to them for migrating to different parts of the world, including Argentina. ", "Since then about 20 families of «White Russians», as they are known locally, maintain their original «peasant» way of life, many of them living a subsistence economy, in Choele Choel in Río Negro Province.", "\n\nIn 1969, Archbishop Leontius (Vasily Konstantinovich Filipovich) came to Buenos Aires. ", "He set about the task of overcoming the split between the Soviet and the monarchist-minded congregations. ", "He died in 1971, and the split was overcome only in the 1990s.", "\n\nThe last wave of emigration coincided with the Perestroika and included Russians who came in search of permanent work and residence in Argentina.", "\n\nThe current ruling bishop of the Argentine and South American dioceses is Archbishop Platon (Vladimir Udovenko).", "\n\nNotable people\nStepan Erzia, sculptor\nVasily Kharlamov, politician\nJorge Remes Lenicov, finance minister\nLola Melnick (in Spanish), dancer\nGerardo Sofovich, businessman and television personality\nCoti Sorokin, singer-songwriter\n\nSee also\n\n Russians\n Russian diaspora\n Argentina–Russia relations\n Immigration to Argentina\n Cathedral of the Most Holy Trinity, Buenos Aires\n\nNotes\n\nReferences\n\nFurther reading\n\nExternal links\nRussian embassy in Buenos Aires about the Russian community in Argentina\nCenter of Russian Science and Culture in Buenos Aires (in Russian and Spanish only)\nArgentine-Russian Chamber of Commerce and Industry (in Russian and Spanish)\n\n \n\nCategory:European Argentine\nArgentina\n \nCategory:Russian diaspora in South America" ]

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